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Large-Scale Financial Risk Management Services. Jan-Ming Ho Research Fellow. Background. Worldwide credit crisis and the credit rating agencies Enron’s bankruptcy in 2001 Lehman Brother’s in 2008 Synthetic CDO backed by RMBS and CDS The Credit Rating Business Protected Oligopoly - PowerPoint PPT Presentation
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Large-Scale Financial Risk Management Services
Jan-Ming HoResearch Fellow
2
Backgroundbull Worldwide credit crisis and the credit rating agencies
ndash Enronrsquos bankruptcy in 2001ndash Lehman Brotherrsquos in 2008
bull Synthetic CDO backed by RMBS and CDSbull The Credit Rating Business
ndash Protected Oligopolybull SEC designation of NRSROs bull Nationally Recognized Statistical Rating Organizations
ndash Issuer-pays business model and Conflict of interestndash Long-term perspective vs up-to-minute assessment
bull Recommendations (eg Lawrence J White 2010)ndash Allowing Wider Choicesndash Bond managerrsquos choice of reliable advisorsndash Prudential oversight of regulators
3
Taking the Opportunity
bull Corporate Credit Ratingbull Computing Risk Measurebull Real-time Derivative Valuation Servicebull Benchmarking Trading Algorithms
Corporate Credit Rating
4
5
Corporate Credit Ratingbull Credit Rating
bull Rating agencies such as Moodys Investors Services and Standard amp Poors (SampP)
bull 21 and 22 classes for long term ratingbull Our method
ndash Using Duffiersquos model to estimate default probability
ndash Optimal partition of default probabilities into classes
6
Duffiersquos Model of Default Probabilitybull Default event
ndash A Poisson process with conditionally deterministic time-varying intensity
bull Default intensity of bankruptcy and other-exit ndash Function of stochastic covariatesndash Firm-specific and macroeconomic
bull Maximum Likelihood Estimation ndash Default probability of a firm in the next quarter
Notations
7
Likelihood Function
8
9
Power curve
bull Cumulative accuracy profile (CAP)bull Sorting the in-sample conditional default
probabilities in non-increasing orderbull Percentage of accumulated defaulted
firms in the next quarter
Power Curve
Perfect Model
A
accuracy ratio (AR) = BA
com
panies defaultedIn the next quarter
Optimal Quantization of Power Curve (OQPC)
The Problem OQPCbull Given a monotonically non-decreasing
array of numbers f[0n]bull Find k cuts ci|1 le i le k ci [0 n] 0 lt cisin 1 lt
c2 lt lt ck lt nbull Such that The area enclosed by the array
C=0c1c2 ck n is maximized
Dynamic Programming
bull The algorithm for DP-QMA runs in O(kn^2) time
Mononiticity of Tail Areas
bull θ(k i) is monotonic increasing in i ie If i ge j then θ(k i) ge θ(k j)
Improved Dynamic Programming
bull The algorithm DP2-QMA runs in O(kn^2) time
Optimal Cuts of Continuous Power Curve
bull If x1 z x2 are 3 consecutive cuts and z is an optimal cut between the cuts x1 and x2 then the fprime(z) must be equal to the slope of AB
Continuous Algorithm
bull This algorithm runs in O(k log^2 n) time
Enclosing Slopes
bull The enclosing slopes of the point C is the slope of the segment AC and the slope of the segment BC
Enclosing Slopes Algorithm
bull Algorithm DC-QMA runs in O(k nlog n) time
Linear Time Heuristic
bull We observed thatΦ+(k n) is a convex function of n Θ+(k n) is monotonic in n and Θ+(k i) ge Θ+(k j) if i gt j
bull If the above claim is true then we have an O(k n) time algorithm
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
3
Taking the Opportunity
bull Corporate Credit Ratingbull Computing Risk Measurebull Real-time Derivative Valuation Servicebull Benchmarking Trading Algorithms
Corporate Credit Rating
4
5
Corporate Credit Ratingbull Credit Rating
bull Rating agencies such as Moodys Investors Services and Standard amp Poors (SampP)
bull 21 and 22 classes for long term ratingbull Our method
ndash Using Duffiersquos model to estimate default probability
ndash Optimal partition of default probabilities into classes
6
Duffiersquos Model of Default Probabilitybull Default event
ndash A Poisson process with conditionally deterministic time-varying intensity
bull Default intensity of bankruptcy and other-exit ndash Function of stochastic covariatesndash Firm-specific and macroeconomic
bull Maximum Likelihood Estimation ndash Default probability of a firm in the next quarter
Notations
7
Likelihood Function
8
9
Power curve
bull Cumulative accuracy profile (CAP)bull Sorting the in-sample conditional default
probabilities in non-increasing orderbull Percentage of accumulated defaulted
firms in the next quarter
Power Curve
Perfect Model
A
accuracy ratio (AR) = BA
com
panies defaultedIn the next quarter
Optimal Quantization of Power Curve (OQPC)
The Problem OQPCbull Given a monotonically non-decreasing
array of numbers f[0n]bull Find k cuts ci|1 le i le k ci [0 n] 0 lt cisin 1 lt
c2 lt lt ck lt nbull Such that The area enclosed by the array
C=0c1c2 ck n is maximized
Dynamic Programming
bull The algorithm for DP-QMA runs in O(kn^2) time
Mononiticity of Tail Areas
bull θ(k i) is monotonic increasing in i ie If i ge j then θ(k i) ge θ(k j)
Improved Dynamic Programming
bull The algorithm DP2-QMA runs in O(kn^2) time
Optimal Cuts of Continuous Power Curve
bull If x1 z x2 are 3 consecutive cuts and z is an optimal cut between the cuts x1 and x2 then the fprime(z) must be equal to the slope of AB
Continuous Algorithm
bull This algorithm runs in O(k log^2 n) time
Enclosing Slopes
bull The enclosing slopes of the point C is the slope of the segment AC and the slope of the segment BC
Enclosing Slopes Algorithm
bull Algorithm DC-QMA runs in O(k nlog n) time
Linear Time Heuristic
bull We observed thatΦ+(k n) is a convex function of n Θ+(k n) is monotonic in n and Θ+(k i) ge Θ+(k j) if i gt j
bull If the above claim is true then we have an O(k n) time algorithm
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Corporate Credit Rating
4
5
Corporate Credit Ratingbull Credit Rating
bull Rating agencies such as Moodys Investors Services and Standard amp Poors (SampP)
bull 21 and 22 classes for long term ratingbull Our method
ndash Using Duffiersquos model to estimate default probability
ndash Optimal partition of default probabilities into classes
6
Duffiersquos Model of Default Probabilitybull Default event
ndash A Poisson process with conditionally deterministic time-varying intensity
bull Default intensity of bankruptcy and other-exit ndash Function of stochastic covariatesndash Firm-specific and macroeconomic
bull Maximum Likelihood Estimation ndash Default probability of a firm in the next quarter
Notations
7
Likelihood Function
8
9
Power curve
bull Cumulative accuracy profile (CAP)bull Sorting the in-sample conditional default
probabilities in non-increasing orderbull Percentage of accumulated defaulted
firms in the next quarter
Power Curve
Perfect Model
A
accuracy ratio (AR) = BA
com
panies defaultedIn the next quarter
Optimal Quantization of Power Curve (OQPC)
The Problem OQPCbull Given a monotonically non-decreasing
array of numbers f[0n]bull Find k cuts ci|1 le i le k ci [0 n] 0 lt cisin 1 lt
c2 lt lt ck lt nbull Such that The area enclosed by the array
C=0c1c2 ck n is maximized
Dynamic Programming
bull The algorithm for DP-QMA runs in O(kn^2) time
Mononiticity of Tail Areas
bull θ(k i) is monotonic increasing in i ie If i ge j then θ(k i) ge θ(k j)
Improved Dynamic Programming
bull The algorithm DP2-QMA runs in O(kn^2) time
Optimal Cuts of Continuous Power Curve
bull If x1 z x2 are 3 consecutive cuts and z is an optimal cut between the cuts x1 and x2 then the fprime(z) must be equal to the slope of AB
Continuous Algorithm
bull This algorithm runs in O(k log^2 n) time
Enclosing Slopes
bull The enclosing slopes of the point C is the slope of the segment AC and the slope of the segment BC
Enclosing Slopes Algorithm
bull Algorithm DC-QMA runs in O(k nlog n) time
Linear Time Heuristic
bull We observed thatΦ+(k n) is a convex function of n Θ+(k n) is monotonic in n and Θ+(k i) ge Θ+(k j) if i gt j
bull If the above claim is true then we have an O(k n) time algorithm
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
5
Corporate Credit Ratingbull Credit Rating
bull Rating agencies such as Moodys Investors Services and Standard amp Poors (SampP)
bull 21 and 22 classes for long term ratingbull Our method
ndash Using Duffiersquos model to estimate default probability
ndash Optimal partition of default probabilities into classes
6
Duffiersquos Model of Default Probabilitybull Default event
ndash A Poisson process with conditionally deterministic time-varying intensity
bull Default intensity of bankruptcy and other-exit ndash Function of stochastic covariatesndash Firm-specific and macroeconomic
bull Maximum Likelihood Estimation ndash Default probability of a firm in the next quarter
Notations
7
Likelihood Function
8
9
Power curve
bull Cumulative accuracy profile (CAP)bull Sorting the in-sample conditional default
probabilities in non-increasing orderbull Percentage of accumulated defaulted
firms in the next quarter
Power Curve
Perfect Model
A
accuracy ratio (AR) = BA
com
panies defaultedIn the next quarter
Optimal Quantization of Power Curve (OQPC)
The Problem OQPCbull Given a monotonically non-decreasing
array of numbers f[0n]bull Find k cuts ci|1 le i le k ci [0 n] 0 lt cisin 1 lt
c2 lt lt ck lt nbull Such that The area enclosed by the array
C=0c1c2 ck n is maximized
Dynamic Programming
bull The algorithm for DP-QMA runs in O(kn^2) time
Mononiticity of Tail Areas
bull θ(k i) is monotonic increasing in i ie If i ge j then θ(k i) ge θ(k j)
Improved Dynamic Programming
bull The algorithm DP2-QMA runs in O(kn^2) time
Optimal Cuts of Continuous Power Curve
bull If x1 z x2 are 3 consecutive cuts and z is an optimal cut between the cuts x1 and x2 then the fprime(z) must be equal to the slope of AB
Continuous Algorithm
bull This algorithm runs in O(k log^2 n) time
Enclosing Slopes
bull The enclosing slopes of the point C is the slope of the segment AC and the slope of the segment BC
Enclosing Slopes Algorithm
bull Algorithm DC-QMA runs in O(k nlog n) time
Linear Time Heuristic
bull We observed thatΦ+(k n) is a convex function of n Θ+(k n) is monotonic in n and Θ+(k i) ge Θ+(k j) if i gt j
bull If the above claim is true then we have an O(k n) time algorithm
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
6
Duffiersquos Model of Default Probabilitybull Default event
ndash A Poisson process with conditionally deterministic time-varying intensity
bull Default intensity of bankruptcy and other-exit ndash Function of stochastic covariatesndash Firm-specific and macroeconomic
bull Maximum Likelihood Estimation ndash Default probability of a firm in the next quarter
Notations
7
Likelihood Function
8
9
Power curve
bull Cumulative accuracy profile (CAP)bull Sorting the in-sample conditional default
probabilities in non-increasing orderbull Percentage of accumulated defaulted
firms in the next quarter
Power Curve
Perfect Model
A
accuracy ratio (AR) = BA
com
panies defaultedIn the next quarter
Optimal Quantization of Power Curve (OQPC)
The Problem OQPCbull Given a monotonically non-decreasing
array of numbers f[0n]bull Find k cuts ci|1 le i le k ci [0 n] 0 lt cisin 1 lt
c2 lt lt ck lt nbull Such that The area enclosed by the array
C=0c1c2 ck n is maximized
Dynamic Programming
bull The algorithm for DP-QMA runs in O(kn^2) time
Mononiticity of Tail Areas
bull θ(k i) is monotonic increasing in i ie If i ge j then θ(k i) ge θ(k j)
Improved Dynamic Programming
bull The algorithm DP2-QMA runs in O(kn^2) time
Optimal Cuts of Continuous Power Curve
bull If x1 z x2 are 3 consecutive cuts and z is an optimal cut between the cuts x1 and x2 then the fprime(z) must be equal to the slope of AB
Continuous Algorithm
bull This algorithm runs in O(k log^2 n) time
Enclosing Slopes
bull The enclosing slopes of the point C is the slope of the segment AC and the slope of the segment BC
Enclosing Slopes Algorithm
bull Algorithm DC-QMA runs in O(k nlog n) time
Linear Time Heuristic
bull We observed thatΦ+(k n) is a convex function of n Θ+(k n) is monotonic in n and Θ+(k i) ge Θ+(k j) if i gt j
bull If the above claim is true then we have an O(k n) time algorithm
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Notations
7
Likelihood Function
8
9
Power curve
bull Cumulative accuracy profile (CAP)bull Sorting the in-sample conditional default
probabilities in non-increasing orderbull Percentage of accumulated defaulted
firms in the next quarter
Power Curve
Perfect Model
A
accuracy ratio (AR) = BA
com
panies defaultedIn the next quarter
Optimal Quantization of Power Curve (OQPC)
The Problem OQPCbull Given a monotonically non-decreasing
array of numbers f[0n]bull Find k cuts ci|1 le i le k ci [0 n] 0 lt cisin 1 lt
c2 lt lt ck lt nbull Such that The area enclosed by the array
C=0c1c2 ck n is maximized
Dynamic Programming
bull The algorithm for DP-QMA runs in O(kn^2) time
Mononiticity of Tail Areas
bull θ(k i) is monotonic increasing in i ie If i ge j then θ(k i) ge θ(k j)
Improved Dynamic Programming
bull The algorithm DP2-QMA runs in O(kn^2) time
Optimal Cuts of Continuous Power Curve
bull If x1 z x2 are 3 consecutive cuts and z is an optimal cut between the cuts x1 and x2 then the fprime(z) must be equal to the slope of AB
Continuous Algorithm
bull This algorithm runs in O(k log^2 n) time
Enclosing Slopes
bull The enclosing slopes of the point C is the slope of the segment AC and the slope of the segment BC
Enclosing Slopes Algorithm
bull Algorithm DC-QMA runs in O(k nlog n) time
Linear Time Heuristic
bull We observed thatΦ+(k n) is a convex function of n Θ+(k n) is monotonic in n and Θ+(k i) ge Θ+(k j) if i gt j
bull If the above claim is true then we have an O(k n) time algorithm
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Likelihood Function
8
9
Power curve
bull Cumulative accuracy profile (CAP)bull Sorting the in-sample conditional default
probabilities in non-increasing orderbull Percentage of accumulated defaulted
firms in the next quarter
Power Curve
Perfect Model
A
accuracy ratio (AR) = BA
com
panies defaultedIn the next quarter
Optimal Quantization of Power Curve (OQPC)
The Problem OQPCbull Given a monotonically non-decreasing
array of numbers f[0n]bull Find k cuts ci|1 le i le k ci [0 n] 0 lt cisin 1 lt
c2 lt lt ck lt nbull Such that The area enclosed by the array
C=0c1c2 ck n is maximized
Dynamic Programming
bull The algorithm for DP-QMA runs in O(kn^2) time
Mononiticity of Tail Areas
bull θ(k i) is monotonic increasing in i ie If i ge j then θ(k i) ge θ(k j)
Improved Dynamic Programming
bull The algorithm DP2-QMA runs in O(kn^2) time
Optimal Cuts of Continuous Power Curve
bull If x1 z x2 are 3 consecutive cuts and z is an optimal cut between the cuts x1 and x2 then the fprime(z) must be equal to the slope of AB
Continuous Algorithm
bull This algorithm runs in O(k log^2 n) time
Enclosing Slopes
bull The enclosing slopes of the point C is the slope of the segment AC and the slope of the segment BC
Enclosing Slopes Algorithm
bull Algorithm DC-QMA runs in O(k nlog n) time
Linear Time Heuristic
bull We observed thatΦ+(k n) is a convex function of n Θ+(k n) is monotonic in n and Θ+(k i) ge Θ+(k j) if i gt j
bull If the above claim is true then we have an O(k n) time algorithm
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
9
Power curve
bull Cumulative accuracy profile (CAP)bull Sorting the in-sample conditional default
probabilities in non-increasing orderbull Percentage of accumulated defaulted
firms in the next quarter
Power Curve
Perfect Model
A
accuracy ratio (AR) = BA
com
panies defaultedIn the next quarter
Optimal Quantization of Power Curve (OQPC)
The Problem OQPCbull Given a monotonically non-decreasing
array of numbers f[0n]bull Find k cuts ci|1 le i le k ci [0 n] 0 lt cisin 1 lt
c2 lt lt ck lt nbull Such that The area enclosed by the array
C=0c1c2 ck n is maximized
Dynamic Programming
bull The algorithm for DP-QMA runs in O(kn^2) time
Mononiticity of Tail Areas
bull θ(k i) is monotonic increasing in i ie If i ge j then θ(k i) ge θ(k j)
Improved Dynamic Programming
bull The algorithm DP2-QMA runs in O(kn^2) time
Optimal Cuts of Continuous Power Curve
bull If x1 z x2 are 3 consecutive cuts and z is an optimal cut between the cuts x1 and x2 then the fprime(z) must be equal to the slope of AB
Continuous Algorithm
bull This algorithm runs in O(k log^2 n) time
Enclosing Slopes
bull The enclosing slopes of the point C is the slope of the segment AC and the slope of the segment BC
Enclosing Slopes Algorithm
bull Algorithm DC-QMA runs in O(k nlog n) time
Linear Time Heuristic
bull We observed thatΦ+(k n) is a convex function of n Θ+(k n) is monotonic in n and Θ+(k i) ge Θ+(k j) if i gt j
bull If the above claim is true then we have an O(k n) time algorithm
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Power Curve
Perfect Model
A
accuracy ratio (AR) = BA
com
panies defaultedIn the next quarter
Optimal Quantization of Power Curve (OQPC)
The Problem OQPCbull Given a monotonically non-decreasing
array of numbers f[0n]bull Find k cuts ci|1 le i le k ci [0 n] 0 lt cisin 1 lt
c2 lt lt ck lt nbull Such that The area enclosed by the array
C=0c1c2 ck n is maximized
Dynamic Programming
bull The algorithm for DP-QMA runs in O(kn^2) time
Mononiticity of Tail Areas
bull θ(k i) is monotonic increasing in i ie If i ge j then θ(k i) ge θ(k j)
Improved Dynamic Programming
bull The algorithm DP2-QMA runs in O(kn^2) time
Optimal Cuts of Continuous Power Curve
bull If x1 z x2 are 3 consecutive cuts and z is an optimal cut between the cuts x1 and x2 then the fprime(z) must be equal to the slope of AB
Continuous Algorithm
bull This algorithm runs in O(k log^2 n) time
Enclosing Slopes
bull The enclosing slopes of the point C is the slope of the segment AC and the slope of the segment BC
Enclosing Slopes Algorithm
bull Algorithm DC-QMA runs in O(k nlog n) time
Linear Time Heuristic
bull We observed thatΦ+(k n) is a convex function of n Θ+(k n) is monotonic in n and Θ+(k i) ge Θ+(k j) if i gt j
bull If the above claim is true then we have an O(k n) time algorithm
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Optimal Quantization of Power Curve (OQPC)
The Problem OQPCbull Given a monotonically non-decreasing
array of numbers f[0n]bull Find k cuts ci|1 le i le k ci [0 n] 0 lt cisin 1 lt
c2 lt lt ck lt nbull Such that The area enclosed by the array
C=0c1c2 ck n is maximized
Dynamic Programming
bull The algorithm for DP-QMA runs in O(kn^2) time
Mononiticity of Tail Areas
bull θ(k i) is monotonic increasing in i ie If i ge j then θ(k i) ge θ(k j)
Improved Dynamic Programming
bull The algorithm DP2-QMA runs in O(kn^2) time
Optimal Cuts of Continuous Power Curve
bull If x1 z x2 are 3 consecutive cuts and z is an optimal cut between the cuts x1 and x2 then the fprime(z) must be equal to the slope of AB
Continuous Algorithm
bull This algorithm runs in O(k log^2 n) time
Enclosing Slopes
bull The enclosing slopes of the point C is the slope of the segment AC and the slope of the segment BC
Enclosing Slopes Algorithm
bull Algorithm DC-QMA runs in O(k nlog n) time
Linear Time Heuristic
bull We observed thatΦ+(k n) is a convex function of n Θ+(k n) is monotonic in n and Θ+(k i) ge Θ+(k j) if i gt j
bull If the above claim is true then we have an O(k n) time algorithm
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
The Problem OQPCbull Given a monotonically non-decreasing
array of numbers f[0n]bull Find k cuts ci|1 le i le k ci [0 n] 0 lt cisin 1 lt
c2 lt lt ck lt nbull Such that The area enclosed by the array
C=0c1c2 ck n is maximized
Dynamic Programming
bull The algorithm for DP-QMA runs in O(kn^2) time
Mononiticity of Tail Areas
bull θ(k i) is monotonic increasing in i ie If i ge j then θ(k i) ge θ(k j)
Improved Dynamic Programming
bull The algorithm DP2-QMA runs in O(kn^2) time
Optimal Cuts of Continuous Power Curve
bull If x1 z x2 are 3 consecutive cuts and z is an optimal cut between the cuts x1 and x2 then the fprime(z) must be equal to the slope of AB
Continuous Algorithm
bull This algorithm runs in O(k log^2 n) time
Enclosing Slopes
bull The enclosing slopes of the point C is the slope of the segment AC and the slope of the segment BC
Enclosing Slopes Algorithm
bull Algorithm DC-QMA runs in O(k nlog n) time
Linear Time Heuristic
bull We observed thatΦ+(k n) is a convex function of n Θ+(k n) is monotonic in n and Θ+(k i) ge Θ+(k j) if i gt j
bull If the above claim is true then we have an O(k n) time algorithm
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Dynamic Programming
bull The algorithm for DP-QMA runs in O(kn^2) time
Mononiticity of Tail Areas
bull θ(k i) is monotonic increasing in i ie If i ge j then θ(k i) ge θ(k j)
Improved Dynamic Programming
bull The algorithm DP2-QMA runs in O(kn^2) time
Optimal Cuts of Continuous Power Curve
bull If x1 z x2 are 3 consecutive cuts and z is an optimal cut between the cuts x1 and x2 then the fprime(z) must be equal to the slope of AB
Continuous Algorithm
bull This algorithm runs in O(k log^2 n) time
Enclosing Slopes
bull The enclosing slopes of the point C is the slope of the segment AC and the slope of the segment BC
Enclosing Slopes Algorithm
bull Algorithm DC-QMA runs in O(k nlog n) time
Linear Time Heuristic
bull We observed thatΦ+(k n) is a convex function of n Θ+(k n) is monotonic in n and Θ+(k i) ge Θ+(k j) if i gt j
bull If the above claim is true then we have an O(k n) time algorithm
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Mononiticity of Tail Areas
bull θ(k i) is monotonic increasing in i ie If i ge j then θ(k i) ge θ(k j)
Improved Dynamic Programming
bull The algorithm DP2-QMA runs in O(kn^2) time
Optimal Cuts of Continuous Power Curve
bull If x1 z x2 are 3 consecutive cuts and z is an optimal cut between the cuts x1 and x2 then the fprime(z) must be equal to the slope of AB
Continuous Algorithm
bull This algorithm runs in O(k log^2 n) time
Enclosing Slopes
bull The enclosing slopes of the point C is the slope of the segment AC and the slope of the segment BC
Enclosing Slopes Algorithm
bull Algorithm DC-QMA runs in O(k nlog n) time
Linear Time Heuristic
bull We observed thatΦ+(k n) is a convex function of n Θ+(k n) is monotonic in n and Θ+(k i) ge Θ+(k j) if i gt j
bull If the above claim is true then we have an O(k n) time algorithm
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Improved Dynamic Programming
bull The algorithm DP2-QMA runs in O(kn^2) time
Optimal Cuts of Continuous Power Curve
bull If x1 z x2 are 3 consecutive cuts and z is an optimal cut between the cuts x1 and x2 then the fprime(z) must be equal to the slope of AB
Continuous Algorithm
bull This algorithm runs in O(k log^2 n) time
Enclosing Slopes
bull The enclosing slopes of the point C is the slope of the segment AC and the slope of the segment BC
Enclosing Slopes Algorithm
bull Algorithm DC-QMA runs in O(k nlog n) time
Linear Time Heuristic
bull We observed thatΦ+(k n) is a convex function of n Θ+(k n) is monotonic in n and Θ+(k i) ge Θ+(k j) if i gt j
bull If the above claim is true then we have an O(k n) time algorithm
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Optimal Cuts of Continuous Power Curve
bull If x1 z x2 are 3 consecutive cuts and z is an optimal cut between the cuts x1 and x2 then the fprime(z) must be equal to the slope of AB
Continuous Algorithm
bull This algorithm runs in O(k log^2 n) time
Enclosing Slopes
bull The enclosing slopes of the point C is the slope of the segment AC and the slope of the segment BC
Enclosing Slopes Algorithm
bull Algorithm DC-QMA runs in O(k nlog n) time
Linear Time Heuristic
bull We observed thatΦ+(k n) is a convex function of n Θ+(k n) is monotonic in n and Θ+(k i) ge Θ+(k j) if i gt j
bull If the above claim is true then we have an O(k n) time algorithm
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Continuous Algorithm
bull This algorithm runs in O(k log^2 n) time
Enclosing Slopes
bull The enclosing slopes of the point C is the slope of the segment AC and the slope of the segment BC
Enclosing Slopes Algorithm
bull Algorithm DC-QMA runs in O(k nlog n) time
Linear Time Heuristic
bull We observed thatΦ+(k n) is a convex function of n Θ+(k n) is monotonic in n and Θ+(k i) ge Θ+(k j) if i gt j
bull If the above claim is true then we have an O(k n) time algorithm
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Enclosing Slopes
bull The enclosing slopes of the point C is the slope of the segment AC and the slope of the segment BC
Enclosing Slopes Algorithm
bull Algorithm DC-QMA runs in O(k nlog n) time
Linear Time Heuristic
bull We observed thatΦ+(k n) is a convex function of n Θ+(k n) is monotonic in n and Θ+(k i) ge Θ+(k j) if i gt j
bull If the above claim is true then we have an O(k n) time algorithm
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Enclosing Slopes Algorithm
bull Algorithm DC-QMA runs in O(k nlog n) time
Linear Time Heuristic
bull We observed thatΦ+(k n) is a convex function of n Θ+(k n) is monotonic in n and Θ+(k i) ge Θ+(k j) if i gt j
bull If the above claim is true then we have an O(k n) time algorithm
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Linear Time Heuristic
bull We observed thatΦ+(k n) is a convex function of n Θ+(k n) is monotonic in n and Θ+(k i) ge Θ+(k j) if i gt j
bull If the above claim is true then we have an O(k n) time algorithm
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
A Linear Time Heuristic
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Numerical Experiment
bull Points sampled from the function
bull Computer environmentndash Pentium Xeon E5630 253G with 70G
memoryndash GCC v461ndash Linux OS
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Running Time ndash Fixed k
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Running Time ndash Fixed n
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
25
Asia Cement
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
26
Real-time Credit Rating
bull Early warning of companies getting close to defaultndash Using real-time market datandash Testing effectiveness and efficiency of subsets
of variables
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Computing Risk Measure
27
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
28
Value at Risk (VaR)bull Early VaR involved along two parallel lines
ndash portfolio theoryndash capital adequacy computations
bull Hardy (1923) and Hicks (1935) ~ non-mathematical discussion of portfolios
bull Leavens (1945) ~ a quantitative examplendash may be the first VaR measure ever published
bull Markowitz (1952) and Roy (1952) ~ a means of selecting portfolios ndash optimize reward for a given level of risk
bull Dusak (1972) ~ simple VaR measures for futures portfolios
bull Lietaer (1971)~ a practical VaR measure for foreign exchange risk ndash integrated a VaR measure with a variance of market value VaR
metric
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
29
bull JP Morgan (1994)ndash Published the extensive development of risk
measurement VaR ndash gave free access to estimates of the necessary
underlying parameters
bull US Securities and Exchange Commission (1997)ndash Major banks and dealers started to implement the rule
that they must disclose quantitative information about their derivatives activities by including VaR information
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
30
Tail conditional expectation (TCE)
bull The tail conditional expectation (TCE) is one of several coherent risk measuresndash P Artzner F Delbaen J-M Eber and D
Heath ldquoCoherent measures of riskrdquo Mathematical Finance vol 9 no 3 pp 203-228 1999
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
31
Definition of Tail Conditional Expectations
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
NT
Probability density
VaR10
-30000 -20000 - 10000 0 10000 20000 30000
Area=10
Value at Risk (VaR)
|P pTCE E V V VaR
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
32
Value of a Sell Put
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
33
Margin Requirement
bull Chicago Board Options Exchange CBOEbull 66 of the margin as collateral
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
34
Modeling Stock Price
bull Lognormal Distributionndash Black ndashScholes
bull Multiplicative Binomial Distribution
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
35
Log-Normal Distribution
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
36
Binomial model
ted
Where
The probability of up is
teu
Here σ is the volatility of the underling stock price and t = one time step time period of σ
dudep
rt
S
Su
Sd
Sd2
Sd3
Sd4
Su2
Sud
Su3
Su4
Su2d
Sud2
Su3d
Su2d2
Sud3
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
37
Expected Value of a Put
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
38
The Problems
bull To speed up the computation of the TCE of a portfolio gain at time T
bull We study two cases ndash Single stock and single option (SSSO) in a
portfoliondash Single stock and multiple options (SSMO) in a
portfolio
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
39
Starting Point
bull We start by computing the TCE of selling a put optionndash Given a put option starting at time t=0 and
strike at maturity time t=U with a strike price Kndash At time t=0 we want to predict the TCE at
time t=T
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
40
Model
bull Given a model of the future price of a stock at time t where 0 t U≦ ≦ndash FS(T) = distribution of stock price S at time Tndash FR (TU) = distribution of price ratio R at time U
with respect to time T where R = FS(U)FS(T)
bull Note that FS and FR can be computed empirically or theoretically
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
41
The SSSO Case
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
42
The SSSO-Naive Algorithm
( )0( ( ))r U T rU
i jv e P e K S R
( )0
r U T rUv e P e
[ | ]i iV E v S
If K ≧ SiRj the portfolio gain (v) equals
If K lt SiRj the portfolio gain (v) equals
The portfolio gain at time T can be computed as follows
bull Under the binomial model selling a put option Vi is strictly decreased when i is increased bull We can determine the position of the p-quantile among the
nodes at time T before calculating the portfolio gain
where P0 is the initial option price i=1hellipm and j=1hellipn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
hellip
helliphellip
hellip
S1
hellip
hellip
hellip
hellip
R1
u
T
S2
S3
Sm
Sm-1
R1
R1
R1
R2
R1
Rn
Rn
Rn
Rn
Rn
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
43
Steps of the SSSO-Naive Algorithm
S1
S2
helliphellip
helliphellip
hellip
r1 = um
S1 = stock_price r1bull The computational
complexity of the SSSO-Naive Algorithm is O(mn)
un
helliphellip
un-1dun-2d2
dnhellip
hellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
helliphellip
un
helliphellip
un-1dun-2d2
dn
p-quantile
helliphellip
Sm-3
Sm-2
Sm-1
Sm
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
44
The SSSO Algorithm
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra1
R1
Rn
Region 2
Region 1
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
S0
S1
helliphellip
helliphellip
helliphellip
helliphellip
Ra2
R1
Rn
Region 1
S2Ra1
Region 3
bull There are two inequalities fromin the binomial model
S1 ge S2 ge hellipgeSm and R1 ge R2 gehellip ge Rn ndash The derived strike price ratio KSi is a
monotonic series KSm ge KSm-1 gehellipge KS1
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
45
The Steps of the SSSO Algorithm
S1
S2
helliphellip
helliphellip
hellip
S3
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
un-2d2
un
un-1d
udn-1
u4dn-4
u3dn-3
u2dn-2
u5dn-5
u6dn-6
dn
bull The computational complexity of the SSSO Algorithm is O(m+n)
p-quantileSm-3
Sm-2
Sm-1
Sm
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
46
Experiment Setting
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
47
bull At time 0 we sell a put option with a strike price K=110 and maturity U=1 year on a stock whose initial price is S0=100
bull The stock price follows the Black-Scholes model ndash normal-distributed drift with μ= 6 and σ= 15 ndash money market account with interest rate r = 6
bull We want to computendash The initial price at which we will sell the put option
P0 ndash TCEp at p=1 level at time T = one week
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
48
Performance Evaluation
where TCEp_SSSO Algorithm is the TCEp value calculated by the SSSO algorithm and TCEp_benchmark is the TCEp value calculated by the Black-Scholes formula
_ lg _
_
100p SSSO A orithm p benchmark
p benchmark
TCE TCEerror rate
TCE
1 accuracy error rate
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
49
Experiment Results of SSSO
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
50
The TCE001 Error Rate Curve of the SSSO Algorithm
0
05
1
15
2
25
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
TCE
Erro
r Rat
e(
)
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
51
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
TCE
Erro
r Rat
e(
)
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
52
The TCE001 Error Rate Curve of the SSSO Algorithm (cont)
0
005
01
015
02
025
03
035
10 60 110 160 210 260 310 360 410 460 510 560 610 660 710 760 810 860 910 960
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
53
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
m (n = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
54
Ratio of the Computation Time of the SSSO-Naive Algorithm to that of the SSSO
Algorithm
0
500
1000
1500
2000
2500
3000
100 600 1100 1600 2100 2600 3100 3600 4100 4600 5100 5600 6100 6600 7100 7600 8100 8600 9100 9600
n (m = 10000 )
Tim
e of
SSS
O-N
aive
Alg
orith
m
Tim
e of
SSS
O A
lgor
ithm
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
55
The TCE0001 Error Rate Curve
000100200300400500600700800901
011012013014015
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
TCE
Erro
r Rat
e(
)
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
56
The Computation Time of the SSSO Algorithm
0
5
10
15
20
25
30
35
40
10 1010 2010 3010 4010 5010 6010 7010 8010 9010
m (10 3 ) (n = 10000 )
Com
puta
tiona
l Tim
e(s
ec)
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
57
Discussion
bull The accuracy of TCE depends primarily on the value m
bull Our algorithm takes less than 5 seconds to compute TCE with 9995 accuracy
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Real-time Derivative Valuation Service
58
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
59
Valuation of Convertible Bonds
Real-time Derivative Valuation Service
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Pricing Convertible Bondsbull The adoption of international financial reporting
standard (IFRS)ndash Banks and Financial Firms ndash Fair value
bull The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date
bull Modeling interest rate and the underlying assetbull Risk associated with convertible bonds
ndash Credit riskndash Market riskndash Liquidity riskndash Optionalities convert call put
60
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Real-time derivative valuation service- User interface (UI)
61
Derivative Profile
Category Derivative Parameters
Derivative Type 1 Derivative type
Derivative Information 1 Valuation date2 Issuing date3 Maturity date4 Credit rating of issuer5 Issue price6 Maturity price7 Country8 Currency9 Coupon rate10 Day count convention11 Holiday setting12 Underlying assets
Derivative Provisions 1 Call provision2 Put provision3 Conversion provision4 Reset provision
Model Setting 1 Valuation model setting2 Stochastic process setting3 Interest model setting4 Correlation method setting
Derivative Type
Callable and Puttable Bonds
Convertible Bonds
Inverse Floaters (Callable)
Capped Floaters (Callable)
Range Accrual Notes (CallablePuttable)
Dual Range Accrual Notes (CallablePuttable)
Index-Linked Notes (CallablePuttable)
Convertible Bond Asset Swap
Range Accrual Swap (Callable)
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Schematic diagram of valuation service
62
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
More Details
63
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
System Architecture
64
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Term sheets of convertible bond
65
European Convertible Bond (ECB)Valuation date 2011325Issue date 2010810Maturity date 2015810Credit rating 2 (TCRI in Taiwan)Country TaiwanCurrency USDIssuing FX rate 318300Issuing conversion price 11076Issue price 100Maturity price 102171Underlying asset The common stock of ACER IncCall provision The issuer can repurchase ECB if the holderrsquos stock price is higher
than the conversion price on 20 consecutive business days after 2013810
Put provision The holder can sell its ECB to the issuer at the put price of 101297 on 2013810
Conversion provision The holder can convert its ECB to common stock at the conversion price between 2010920 and 2015931
Asset SwapDelivery date 2011325Maturity date 2013810Contract 1Receive $100 and pay an American CB call option on the delivery
date2Receive USD 3M LIBOR + 115bps (30360) quarterly3Pay $101297 to counterparty on the maturity date
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Experiments on valuation of convertible bonds
66
bull Benchmarkndash Market prices as benchmarkndash Fair value is not market price
bull Parameter settingndash Underlying asset simulation
bull Stock price and exchange ratesndash Geometric Brownian motion
bull Risk-free interest ratesndash CIR interest rate model
bull Volatility calculationndash Historical volatility
bull Simulating 2000000 paths for each underlying assetndash Valuation model
bull Least-Square Monte-Carlo method with antithetic variables
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Experiments on valuation of convertible bond
67
bull European convertible bond (ECB) and asset swap (CBAS)
ECB market price 1049287 1041787 Valuation date 2011325Fair value of ECB 104378671Standard error 0036345
Fair value of CBAS 0004817Fair value of CB option in CBAS 6799198Fair value of synthetic straight bond in CBAS 97579473
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Real-time derivative valuation service- Illustration of valuation results
68
CB ID 23171 (TCRI rating 4) CB ID 19023 (TCRI rating 5)
CB ID 14773 (TCRI rating 3) CB ID 140201 (TCRI rating 4)
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Real-time derivative valuation service- Experiment of convertible bond valuation
69
bull 46 convertible bonds (CBs) in Taiwan marketndash Monthly valuation from 2006 to 2010
bull Evaluation metricsndash Normalized RMSE
bull mi is the market price on day i bull vi is the fair value on day i
ndash Maximum of Relative Absolute Error
bull mi is the market price of the i-th daybull vi is the fair value of the i-th day
i
iini m
mvMax 1
n
i i
n
i ii
m
mv
1
21
2
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Real-time derivative valuation service- Experiment of convertible bond valuation
CB ID NRMSEMax of Relative Absolute
Error
Fair Value Market PriceNumber of
ObservationsTCRI
RatingMean Variance Mean Variance
12101 0138 0370 115198 68248 103926 67882 40 4
12161 0093 0225 105716 33195 98818 7505 37 3
12251 0055 0114 117511 49358 114196 9002 13 7
13121 0093 0246 117709 346118 110227 333042 31 4
140201 0107 0200 107274 79305 99562 16486 37 4
14093 0106 0279 118812 276655 108979 328398 45 5
14371 0078 0105 247213 1707347 233800 1222200 5 6
14772 0025 0061 223617 1585860 221571 1609648 14 3
14773 0036 0136 153428 354294 149976 451527 19 3
15221 0058 0139 111222 122354 107597 110457 30 5
15242 0126 0154 98512 6172 112500 1000 4 7
15281 0092 0184 103171 290304 101296 114455 25 6
15291 0077 0126 144889 1416439 137543 1293046 7 6
15323 0045 0104 287122 786220 288000 583000 7 7
15371 0063 0162 134457 1300202 129312 1549028 33 570
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Stock Model Geometric Brownian motion (GBM)
ndash St Stock price at time tndash Wt Wiener process or Brownian motionndash μ Drift term is constantndash σ Volatility of stock prices is constant
tttt dWSdtSdS
Source Wikipedia
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Modified least-square Monte-Carlo model
72
bull Properties of geometric Brownian motionndash Period from pricing date is longer variations of reference entities
is largerbull Keeping less information near the pricing date and
sampling more paths when near the maturity date
Stock Price Simulation by geometric Brownian motion
(100 paths are selected randomly)
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Credit risk
bull Estimating default intensity (hazard rate) using Duffiersquos model
ratehazardratereceveryspreadyield 1
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Interest rate Modelbull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Market riskbull Equity risk
ndash Using historical volatility calculation to model stock prices
bull Ri is the rate of return at time i and n is the calculation period
bull Interest rate riskndash Simulating interest rates by using interest rate model with observable data in the market
bull CIR interest rate model (follows mean-reverting process)
ndash Ensuring mean reversion of the interest rate towards the long run value b with speed of adjustment governed by the strictly positive parameter a
ndash σ is a volatility of interest rate and Wt is a Wiener processbull LIBOR market model
ndash Lj is the forward rate for the period [Tj Tj+1] and σj is a volatility of forward rate for the period [Tj Tj+1]
ndash WQTj is a Wiener process under Tj-forward measure QTj
tttt dWrdtrbadr
)()()()( tdWtLttdL jTQjjj
2521)(
1
2
n
ii
nRR
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Liquidity risk - 1bull Definition
ndash The gap between fundamental value and actually transacted value
bull Traditional estimation approachndash Information cost model
bull Trading volumebull Bid-ask spreads
ndash Problembull Measurement is unavailable when the derivative is illiquid
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Liquidity risk -2bull Latent liquidity
ndash Measuring liquidity of a derivative without using transaction datandash It measures the accessibility of a security from sources where the
security is currently being held
bull Lit is latent liquidity for bond I at time t
bull πijt is the fractional holding of fund j at the end of month t
bull Tjt is the turnover of fund j from month t to month t-12
bull Estimating latent liquidity of a security by using its propertyndash Credit quality age issue size and optionalities such as call put or convertibility
j tj
itj
it TL
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Liquidity Risk - 3
bull Valuejt is the value of fund j at the end of month t
bull Voljt is the dollar trading volume of fund j from month t to month t-12
tj
tjtj Vol
ValueT
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
79
VALUATION OF MORTGAGESReal-time Derivative Valuation Service
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
80
Mortgage and Its Derivatives
bull RMBS and CDObull Valuation of Mortgage
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
81
Pricing Mortgage Loanbull Fixed-Rate Mortgages (FRMs)bull Default Risk
ndash Borrower unwilling or unable to pay their debtndash Auction off security to redeem partial amount of
moneybull Prepayment Risk
ndash Refinancingndash Receive full amount of the outstanding ndash Lose all interest after the prepayment date
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Fixed-Rate Mortgages (FRMs)bull Fully amortizing
ndash Constant interest ratendash Constant payment
bull Value of the FRM to the bankbull The Ideal Cash Flow
82
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Constant payment
Constant payment
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Default bull Borrower unwilling or unable to pay their debt
ndash Partial amount of moneybull Auction security
83
Partial amount of money
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Default date
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Prepayment bull Debt is repaid in advanced
ndash Fully prepaymentbull Refinancingbull Receive all of the outstanding bull Lose all interest after the prepayment date
84
Outstanding
Lending date
principal
Payment date 1 Payment date 2 Payment date n-1 Maturity date
Constant payment
Constant payment
Constant payment
Prepayment date
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Weaknesses of the Previous Approachbull Prepayment loss rate
85
1 12 23 34 45 56 67 78 89 1001111221331441551661771881992102212320
001002003004005006007008009
01Prepayment Loss Rate
PaymentDate
Loss Rate
outstandingprepayment loss rate=1-mortgage price
Determined
Using Tsai el al(2009) model
Not a constant
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
86
Current Progress
bull Modeling and derivation of exact solution of valuating mortgages
bull Opera Solutions
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
87
Modeling Mortgage Loan
bull Default and prepayment riskndash Poisson processes with time-varying intensitiesndash Interest rate as the only state variablendash Intensities as linear functions of interest rate
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
bull A FRM with fixed payment Y and coupon rate c initial outstanding M0 and maturity T ndash The payment Y
ndash The outstanding principal at time t
bull Discrete approximationndash Risk-neutral pricing model
where Vi is the mortgage price at time i i=012hellipn=TΔt PV() and Ei() denotes the present value and expectation of the information at time i under risk-neutral measure
88
Pricing Framework
)))exp(1((0 cTcMY
)))exp(1()))(exp(1((0 cTtTcMM t
)]([ 11 iiii VYPVEV
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Backward Induction
89
0nV
1nV
2nV
3nV
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Discrete Model with Risk Considerationbull Denote and as default and prepayment
probabilities respectively
with initial probability
with initial probability bull where and denote the time occurrence of default
and prepayment respectivelybull Discrete time model
90
DtP P
tP
default of rate loss theis where)])(1)(exp(
))(exp()1[(E
1
111111
11111
i
iiiiD
iiP
i
iiiD
iP
iii
VtYtrPMP
VtYtrPPV
)|( tttttPP DDDt
)|( tttttPP PPPt
D P
0DtP
0PtP
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Expression of Initial Mortgage Value bull Discrete time form
ndash Backward Induction
bull Continuous time formndash Default and prepayment risk follow Poisson process with
time-varying intensities
91
0 0 01 1
01 1
E [ exp( ( ( (1 ))))] (1 exp( ))
E [ (1 exp( ( ( 1) ))) exp( ( ( (1 )))
( (1 )))]
n iP P D
j j j j j j ji j
n iP P P D
i j j j j j j ji j
P P Di i i i i i i
V Y t r t P r tP P r t M cT
c T i t P r t P r tP P r t
r t P r tP P r t
0 00 0
0 00 0
E [exp( ( ) )]
(1 exp( )) E [(1 exp( ( ))) exp( ( ) )]
T t P Du u u
T tP P Dt u u u
V Y r du dt
M cT c T t r du dt
First expectation
Second expectation
lyrespective and Pt
Dt
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Closed-Form Formula
92
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
93
Benchmarking Trading Algorithms
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
94
Benchmarking Trading Algorithms
bull Mutual fund and trading algorithmsbull Maximum return subject to number of
transactionsndash the all-in-all-out trading strategy
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
95
20030902200404022004102820050531200512232006072720070301200709262008042820081120200906232010011320100813201103154000
5000
6000
7000
8000
9000
10000
20030901-20110831 Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)
Series1
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
96
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2003-2004
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2004-2005
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2005-2006
1 22 43 64 85 106127148169190211232400050006000700080009000
10000
2006-2007
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2007-2008
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2008-2009
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2009-2010
1 22 43 64 85 1061271481691902112324000
5000
6000
7000
8000
9000
10000
2010-2011
Sept 1 to Aug 30 ofthe next year
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
97
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 670
50
100
150
200
250
Max Return st Trades Constraint
200309-200408200409-200508200509-200608200609-200708200709-200808200809-200908200909-201008201009-201108
Number of Trades
Retu
rn ra
tio T
rans
actio
n fe
e ad
just
ed
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
98200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
0
50
100
150
200
250
0
0005
001
0015
002
0025
Volatility of Return vs Max Return of Top-N Trades
Max return of top 1 tradesMax return of top 10 tradesMax return of top 20 tradesMax return of top 30 tradesMax return of top 40 tradesMax return of top 50 tradesMax return of top 60 tradesMax return without constraintvolatility of returnRe
turn
Vola
tility
- th
e bl
ue(to
p) li
ne o
nly
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
99
200309-200408
200409-200508
200509-200608
200609-200708
200709-200808
200809-200908
200909-201008
201009-201108
-10000
-5000
000
5000
10000
15000
20000
25000
MRT vs Return of Top Mutual Funds (Each Year)
日盛小而美統一黑馬台新科技 ( 台新台新 )永豐中小兆豐國際豐台灣柏瑞巨人統一大滿貫Max return of top-1 tradeMax return of top-10 tradesMax return without constraint
Retu
rn
K=10
K=1
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
100
Research Team
bull Prof William WY Hsubull Dr Cheng-Yu Lubull Yi-Cheng Tsaibull Da-Wei Hungbull Hsin-Tsung Peng
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
Collaboratorsbull Chung-Su Wu President of CIERbull Ming-Yang Kao Department of EECS
Northwestern Universitybull Yuh-Dau Lu Department of CSIE NTUbull Szu-Lang Liao Department of Money and
Banking NCCUbull Tai-Chang Wang Department of Accounting NTUbull Ruey S Tsay University of Chicago bull Jin-Chuan Duan Director of Risk Management
Institute NUS
101
