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Zvi Wiener

mswiener@mscc.huji.ac.il

02-588-3049

Financial Engineering

FE-Whttp://pluto.mscc.huji.ac.il/

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Following

Paul Wilmott, Introduces Quantitative Finance

Chapter 7

Elementary Stochastic Calculus

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 3

Coin Tossing

Ri = -1 or 1 with probability 50%

E[Ri] = 0

E[Ri2] = 1

E[Ri Rj] = 0

Define

j

iij RS

1

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 4

Coin Tossing

0jSE

jRRRESE j 212

12

5516 ,, SRRSE

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 5

Markov Property

No memory except of the current state.

Transition matrix defines the whole dynamic.

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 6

The Martingale Property

Some technical conditions are required as well.

jji SijSSE ,

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 7

Quadratic Variation

For example of a fair coin toss it is = i

i

jjj SS

1

21

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 8

Brownian Motion

0)( tSE

ttSE 2)(

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 9

Brownian Motion

Finiteness – does not diverge

Continuity

Markov

Martingale

Quadratic variation is t

Normality: X(ti) – X(ti-1) ~ N(0, ti-ti-1)

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 10

Stochastic Integration

n

jjjj

t

ntXtXtfdXftW

111

0

)()()(lim)()()(

n

tjt j

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 11

Stochastic Differential Equations

t

dXftW0

)()()(

dXtfdw )(

tdX has 0 mean and standard deviation

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 12

Stochastic Differential Equations

dXtfdttgdw )()(

tt

dXfdgtW00

)()()()(

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 13

Simulating Markov Process

The Wiener process ),0( tNX

The Generalized Wiener process

XbtaS

The Ito process

XtSbttSaS ),(),(

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 14

time

value

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 15

Ito’s Lemma

dt dX

dt 0 0

dX 0 dt

dtdX

FddX

dX

dFdF

tXF

2

2

2

1

),(

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 16

Arithmetic Brownian Motion

At time 0 we know that S(t) is distributed

normally with mean S(0)+t and variance 2t.

dXdtdS )0()()0()( XtXtStS

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 17

Arithmetic BM dS = dt + dX

time

S

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 18

The Geometric Brownian Motion

Used for stock prices, exchange rates.

is the expected price appreciation:

= total - q.

S follows a lognormal distribution.

XStSS

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 19

The Geometric Brownian MotiondXSdtSdS

SLogF

dXdtdF

2

2

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 20

The Geometric Brownian Motion

)0()(

2)0()(

2

XtXtExpStS

dtS

VSdS

S

Vdt

t

VdV

2

222

2

1

),( tSV

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 21

Geometric BM dS = Sdt + SdX

time

S

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 22

The Geometric Brownian Motion

)1,0(

2

2lnln NtT SS

ttNS

S

2,~

ttSSS Nttt )1,0(1

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 23

Mean-Reverting Processes

dXdtSdS

dXSdtSdS

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 24

Mean-Reverting Processes

dXSdtSdS

SF

dXdtF

FdF

228

4 2

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 25

Simulating Yields

GBM processes are widely used for stock prices

and currencies (not interest rates). A typical

model of interest rates dynamics:

tttt zrtrbkr )(

Speed of mean reversion Long term mean

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 26

Simulating Yields

= 0 - Vasicek model, changes are normally

distr.

= 1 - lognormal model, RiskMetrics.

= 0.5 - Cox, Ingersoll, Ross model (CIR).

tttt zrtrbkr )(

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 27

Mean Reverting Process

dS = (-S)dt + SdX

time

S

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 28

Other models

Ho-Lee term-structure model

HJM (Heath, Jarrow, Morton) is based on forward

rates - no-arbitrage type.

Hull-White model:

tt zttr )(

ttt ztartr )(

Zvi Wiener FE-Wilmott-IntroQF Ch7 slide 29

Home AssignmentRead chapter 7 in Wilmott.

Follow Excel files coming with the book.