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IS-1 Financial PrimerStochastic Modeling
SymposiumBy
Thomas S.Y. Ho PhDThomas Ho Company, [email protected]
April 3, 2006
2
Purpose Overview of the basic principles in the
relative valuation models Overview of the basic terminologies
Equity derivatives Fixed income securities
Practical implementation of the models Examples of applications
3
“Traditional Valuation” Net present value Expected cashflows Cost of capital as opposed to cost of
funding Capital asset pricing model Cost of capital of a firm as opposed to cost
of capital of a project (or security)
4
Relative Valuation Law of one price: extending to non-
tradable financial instruments Applicability to insurance products and
annuities (loans and GICs) Arbitrage process and relative pricing
5
Stock Option Model Modeling approach: specifying the
assumptions, types of assumptions Description of an option Economic assumptions:
Constant risk free rate Constant volatility Stock return distribution Efficient capital markets
6
Binomial Lattice Model Generality of the model in describing the
equity return distribution Market lattice and risk neutral lattice Dynamic hedging and valuation Intuitive explanation of the model results Comparing the relative valuation approach
and the traditional approach – the case of a long dated equity put option
7
One-Period Binomial Model Su/S > exp(rT)> Sd/S In the absence of arbitrage opportunities, there
exist positive state prices such that the price of any security is the sum across the states of the world of its payoff multiplied by the state price.
=(Cu – Cd)/(Su -Sd )
Πu =(S- exp(-rT) Sd )/(Su - Sd )
C = πuCu + πdCd
S= πuSu + πdSd
1 = πuexp(rT)+ πdexp(rT)
8
Numerical Example: Call Option Pricing
Stock Price($) S 100
Strike Price ($) X 100
Stock Volatility σS 0.2
Time to expiration (year) T 1
Risk-free rate r 0.05
dividend yields d N/A
the number of periods n 6 dt = T/n
upward movement u 1.0851 = exp(σ√dt)
downward movement d 0.9216 = 1/u
risk-neutral probability of u p 0.5308 = (exp(rdt)-d)/(u-d)
9
Stock lattice 163.214965
150.418059
138.624497
138.624497
127.755612
117.738905
127.755612
117.738905
108.50756
100
117.738905
108.50756
10092.1594775
84.933693
108.50756
10092.1594775
84.933693
78.2744477
72.1373221
stock lattice 10092.1594775
84.933693
78.2744477
72.1373221
66.4813791
61.2688917
time 0 1 2 3 4 5 6
10
Call Option Lattice 63.214965
51.247930 38.624497
40.277352 28.585483 17.738905
30.224621 19.391759 9.337430 0.000000
21.723634 12.494533 4.915050 0.000000 0.000000
15.055460 7.780762 2.587191 0.000000 0.000000 0.000000
10.125573 4.729344 1.361849 0.000000 0.000000 0.000000 0.000000
11
Martingale Processes, p and q measures C/R = puCu/Ru + pdCd/Rd
S/R = puSu/Ru + pdSd/Rd
1 = pu + pd
C/S = quCu/Su + qdCd/Sd
R/S = quRu/Su + qdRd/Sd
1 = qu + qd
Probability measure: assigning prob Denominator: numeraire Martingale: “expected” value= current value
12
Continuous Time Modeling Ito process
dX(t) = µ(t)dt + σ(t)dB(t) (dt)2 =0 (dt)(dB)=0 (dB)2 =dt
Z = g( t, X) dZ = gt dt + gXdX + 1/2 gxx (dX)2
Geometric Brownian motion dS/S =µdt + σdB(t) S(t) = S(0)exp (µt - σ2t/2 + σ B(t))
13
Numeraires and Probabilities dS/S = µs dt + σsdBs(t) dividend paying dV/V = qdt + dS/S dividend re-invested dY/Y = µ* dt + σ*dB*(t) any asset R(t) = integral of r(s) stochastic rates Risk neutral measure
Z(t) = V(t)/R(t) dS/S = (r- q) dt + σsdB(t)
V as numeraire Z(t) = R(t)/V(t) dS/S = (r – q + σs
2)dt + σs dB’
14
Numeraire General Case Y as numeraire
Z(t) = V(t)/Y(t) dS/S = (r – q + ρσs σy)dt + σs dB’’
Volatility invariant
15
Risk Neutral Measure Martingale process Examples of measures
p measure, forward measure, market measure Generalization of the Black-Scholes Model Applications in the capital markets Applications to the insurance products
Life products Fixed annuities Variable annuities
16
Sensitivity Measures Delta , S Gamma Г, Theta θ (time decay) t Vega v measure σ Rho , r Relationships of the sensitivity measures Intuitive explanation of the greeks
European, American, Bermudian, Asian put/call options Comparing with the equilibrium models
Continual adjustment of the implied volatility
17
??
Stock Price (S) 100
Strike Price (K) 100
Time to expiration (T) 1
Stock volitility (σ) 0.2
Risk-free rate (r) 0.04
Dividend yields (δ) 0
18
Numerical Example of the Greeks
Call Put
Price 9.92505 6.00400
Δ(Delta) 0.61791 -0.38209
Γ(Gamma) 0.01907 0.01907
v (Vega) 38.13878 38.13878
Θ(Theta) -5.88852 -2.04536
ρ (Rho) 51.86609 -44.21286
19
Interest Rate Modeling Lattice models Yield curve estimation Yield curve movements Dynamic hedging of bonds Term structure of volatilities Sensitivity measures
Duration, key rate duration, convexity
20
Interest Rate Model: Setting Up
year 0 1 2 3 4 5
initial yield curve 0.060 0.060 0.065 0.070 0.075 0.080
initial discount function p(n)1.00000
00.94176
50.87809
50.81058
40.74081
80.67032
0
one period forward curve 0.060 0.060 0.070 0.080 0.090 0.100
lognormal spot volatility (σS) 0 0.0775 0.0775 0.0775 0.0775 0.0775
lognormal forward volatility (σf) 0 0.0775 0.0775 0.0775 0.0775 0.0775
21
Ho –Lee (basic) Model
( 1)(1) 2
( ) (1 )
ini n
P nP
P n
0.86124
1
0.87964
7 0.87469
5
0.89695
1 0.89200
4 0.88835
8
0.91310
5 0.90814
3 0.90453
4 0.90223
5
0.92805
8 0.92306
6 0.91947
4 0.91724
1 0.91632
8
Discount function lattice0.94176
5 0.93672
9 0.93313
6 0.93094
6 0.93012
6 0.93064
2
year 0 1 2 3 4 5
22
Ho-Lee One Period Rates
ln ( )( )
nn ii
P Tr T
T
0.14938
06
0.12823
510.13388
06
0.10875
380.11428
510.11838
06
0.09090
470.09635
380.10033
510.10288
06
0.07466
080.08005
470.08395
380.08638
510.08738
06
Interest rate lattice 0.060.06536
080.06920
470.07155
380.07243
510.07188
06
year 0 1 2 3 4 5
23
1 2 1 1
1 2
1 1 2( 1)(1)
( ) 1 1 1n n nn i
i nn n n
P nP
P n
0.86673
1
0.88096
3 0.87807
2
0.89598
0 0.89266
8 0.88956
2
0.911395
0.907795
0.904530
0.901201
0.926800
0.923044
0.919765
0.916549
0.912994
Discount function lattice0.94176
5 0.937988
0.934841
0.931893
0.928727
0.924940
year 0 1 2 3 4 5
24
1 2 1 1
1 2
1 1 2( 1)(1)
( ) 1 1 1n n nn i
i nn n n
P nP
P n
Ho-Lee model rates
with term structure of volatilities
0.1430264
0.1267402 0.1300264
0.1098368 0.1135402 0.1170264
0.0927784 0.0967368 0.1003402 0.1040264
0.076018 0.0800784 0.0836368 0.0871402 0.0910264
0.06 0.064018 0.0673784 0.0705368 0.0739402 0.0780264
0 1 2 3 4 5
lognormal spot volatility (σS) 0 0.1 0.095 0.09 0.085 0.08
lognormal forward volatility (σf) 0 0.1 0.0907143 0.081875 0.0733333 0.065
1
25
Alternative Arbitrage-free Interest Rate Modeling Techniques These are not economic models but
techniques Spot rate model N-factor model Lattice model Continuous time model Calibrations
26
Alternative Valuation Algorithms Discounting along the spot curve Backward substitution Pathwise valuation
monte-carlo Antithetic, control variate Structured sampling
Finite difference methods
27
Example of Interest Rate Models Ho-Lee, Black-Derman-Toy, Hull-White Heath-Jarrow-Morton model Brace-Gatarek-Musiela/Jamshidian model (Market Model) String model Affine model
28
Examples of Applications Corporate bonds (liquidity and credit risks)
Option adjusted spreads Mortgage-backed securities
Prepayment models CMOs
Capital structure arbitrage valuation Insurance products
29
Conclusions Comparing relative valuation and the NPV
model Imagine the world without relative
valuation Beyond the Primer:
Importance of financial engineering Identifying the economics of the models
30
References Ho and Lee (2005) The Oxford Guide to
Financial Modeling Oxford University Press Excel models (185 models)
www.thomasho.com Email: [email protected]