Andrey Itkin, Math 612-02 Selected Topics in Applied Mathematics – Computational Finance Andrey Itkin itkin Course web page itkin/CompFinanceC

Andrey Itkin, Math 612-02

Selected Topics in Applied Mathematics – Computational

Finance

Andrey Itkin

http://www.chem.ucla.edu/~itkin

Course web page

http://www.chem.ucla.edu/~itkin/CompFinanceCourse/rutgers_course.html

My email: [email protected]

http://www.chem.ucla.edu/~itkin




What is computational finance?

• Why computational?– New sophisticated models– Performance issue– Calibration– Data issue and historical data

• Market demand for quant people• Pre-requisites:

– Stochastic calculus and related math– Financial models– Numerical methods– Programming

Result: CF - very complex subject


Course outline?1. Closed-form solutions (BS world, stochastic volatility and Heston

world, interest rates and Vasichek and Hull-White world)2. Almost closed-form solutions – FFT, Laplace transform3. Traditional probabilistic solutions – binomial, trinomial and

implied trees4. Modern solutions – finite-difference5. Last chance - world of Monte Carlo, stochastic integration6. Calibration – gradient optimizers, Levenberg-Marquardt7. Advanced optimization – pattern search8. Specificity of various financial instruments – exotics, variance

products, complex payoffs.9. Programming issues: Design of financial software, Excel/VBA-C++

bridge, Matlab-C++ bridge10. Levy processes, VG, SSM.


(Numerical technique)

Engineering

Mathematics

(Basic stochastic calculus)

Finance

(Derivative pricing And hedging)

This course

Excerpt from Yuji Yamada’s course


Lecture 1

1. Short overview of stochastic calculus

2. All we have to know about Black-Scholes

3. Traditional approach – binomial trees


Binomial Trees

• Binomial trees are used to approximate the movements in the price of a stock or other asset

• In each small interval of time the stock price is assumed to move up by a proportional amount u or to move down by a proportional amount d


Movements in Time t

Su

Sd

S

p

1 – p


Equation of tree Parameters

• We choose the tree parameters p, u, and d so that the tree gives correct values for the mean & standard deviation of the stock price changes in a risk-neutral world

(from John Hull: ) – the expected value of the stock price E(Q) = Ser t = pSu + (1– p ) Sd

Log-normal process: var = S2e2rt(eσ2 t -1) = E(Q2) – [E(Q)]2

2t = pu 2 + (1– p )d 2 – [pu + (1– p )d ]2


Solution to equations

2 equations, 3 unknown. One free choice:

tr

t

t

ea

du

dap

ed

eu

d

u1

Cox Ross Rubinstein

(CRR)


Alternative Solution

• By Jarrow and Rudd

2

1

)2/(

)2/(

2

2

tr

ttr

ttr

ea

du

dap

ed

eu


Pro and Contra

• CRR – it leads to negative probabilities when σ < |(r-q)√t|.

• Jarrow and Rudd – not as easy to calculate gamma and rho.

• If many time steps are chosen – low performance


An alternative exlanations. Single period binomial model (Excerpt from Yuji Yamada’s course)

X1(uS)=uS+r)

X1(dS)=dS+r)

t=1uS

dS

p

1-p

t=0

SStock

(1+r)

(1+r)d<1+r<u

Bond

X0=S+

Portfolio


C0

Single period binomial model

)0,max()(1 KuSuSC

)0,max()(1 KdSdSC

• Compare with portfolio process

)1()()( 11 ruSuSXuSC )1()()( 11 rdSdSXdSC

Two equations fortwo unknowns

Solve these equations for and


)1()(1 ruSuSC

)1()(1 rdSdSC

dSuS

dSCuSC

)()( 11

))(1(

)()( 11

dur

dSdCuSuC

SXC 00

Comparison principle

)(

)1()(

1

1

1110 dSC

du

ruuSC

du

dr

rC

11 XC for each state


q~p~

)(

)1()(

1

1

1110 dSC

du

ruuSC

du

dr

rC

)(~)(~1

1110 dSCquSCp

rC

0~,0~,1~~ qpqp

It is (notationally) convenient to regard p~ q~and as probabilities

: Risk neutral probability (real probability is irrelevant)qp ~,~


Multi-period binomial lattice model

S

uS

dS

Su2

udS

Sd 2

Su3

Sd 3

Sud 2

dSu2

Su4

Sd 4

dSu3

Sdu 22

Sud 3

Stock price

KSu 4

0

KdSu 3

KSdu 22

0

Call price

Finite number of one step models


Backwards Induction

• We know the value of the option at the final nodes

• We work back through the tree using risk-neutral valuation to calculate the value of the option at each node, testing for early exercise when appropriate


)0,max()( 334 KuSuSC

3S

3uS

3dS

Stock price

)0,max()( 334 KdSdSC

Call price

)( 33 SC

p~

q~

Apply one step pricing formula at each step, and solve backward until initial price is obtained.

,)(~)(~1

1)( 343433 dSCquSCp

rSC

pq

du

drp ~1~,

1~

11

111

)()(

kk

kkkkk dSuS

dSCuSC )(~)(~1

1)( 1111

kkkkkk dSCquSCp

rSC


• Perfect replication is possible

• Real probability is irrelevant

Market is complete

• Risk neutral probability dominates the pricing formula

Multi-period binomial lattice model


Binomial Trees and Option Pricing(Two Fundamental Theorem of Asset Pricing (FTAP))

• 1st : The no-arbitrage assumption implies there exists (at least) a probability measure Q called risk-neutral, or risk-adjusted, or equivalent martingale measure, under which the discounted prices are martingales

• 2nd: Assuming complete market and no-arbitrage: there exists a unique risk-adjusted probability measure Q; any contingent claim has a unique price that is the discounted Q-expectation of its final pay-off


Binomial Trees and Option Pricing(Cox-Ross-Rubinstein Formula)

• Cox-Ross-Rubinstein Formula:

• J is the set of integers between 0 and N:

• Risk-neutral probability:


Binomial Trees and Option Pricing(Summary)


American Put Option

S0 = 50; X = 50; r =10%; = 40%;

T = 5 months = 0.4167;

t = 1 month = 0.0833

The parameters imply

u = 1.1224; d = 0.8909;

a = 1.0084; p = 0.5076


Example (continued)Figure 18.3

89.070.00

79.350.00

70.70 70.700.00 0.00

62.99 62.990.64 0.00

56.12 56.12 56.122.16 1.30 0.00

50.00 50.00 50.004.49 3.77 2.66

44.55 44.55 44.556.96 6.38 5.45

39.69 39.6910.36 10.31

35.36 35.3614.64 14.64

31.5018.50

28.0721.93

10.3614.64)0.49276.38(0.5073 :Valueon Continuati

3110693950:ExEarly 0833.01.0

e

. . -


Calculation of Delta

Delta is calculated from the nodes at time t

41.055.4412.56

96.616.2


Calculation of Gamma

Gamma is calculated from the nodes at time 2t

)69.3999.62(5.06511

03.065.11

=Gamma

64.069.3950

36.1077.3;24.0

5099.62

77.364.0

21

21

.


Calculation of Theta

Theta is calculated from the central nodes at times 0 and 2t

day calendar per .or

yearper =Theta

0120

3.41667.0

49.477.3

-


Calculation of Vega

• We can proceed as follows

• Construct a new tree with a volatility of 41% instead of 40%.

• Value of option is 4.62

• Vega is

4 62 4 49 013. . . per 1% change in volatility


Trinomial Tree (Hull P.409)Again we want to match the mean and standard deviation

of price changes. Terms of higher order than t are ignored

6

1

212

3

2

6

1

212

/1

2

2

2

2

3

rt

p

p

rt

p

udeu

d

m

u

t

S S

Sd

Su

pu

pm

pd

Equivalent to explicit FD of 1st order


Alternative solutions:

• Combine two steps of CRR:

dum

tt

trt

d

tt

ttr

u

tt

ppp

ee

eep

ee

eep

edeu

1

,

2

22

22

2

22

22

22

Documents

Andrey Itkin, Math 612-02 Selected Topics in Applied Mathematics – Computational Finance Andrey Itkin itkin Course web page itkin/CompFinanceC