CH12- WIENER PROCESSES AND ITÔ'S LEMMA

OUTLINEOUTLINE

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the value of the variable changes only at certain fixed time point

only limited values are possible for the variable

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12.1 THE MARKOV 12.1 THE MARKOV PROPERTY PROPERTY

A Markov process is a particular type of stochastic process .

The past history of the variable and the way that the present has emerged from the past are irrelevant.

A Markov process for stock prices is consistent with weak-form market efficiency.

where only the present value of a variable is relevant for predicting the future.

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12.212.2CONTINUOUS-TIME CONTINUOUS-TIME STOCHASTIC STOCHASTIC PROCESSES PROCESSES

Suppose $10(now), change in its value during 1 year is (,1)

What is the probability distribution of the stock price at the end of 2 years? (,2)

6 months? (,0.5) 3 months? (,0.25) t years? (, t)

N~(μ=0, σ=1)

N~(μ=0, σ=2)

•In Markov processes changes in successive periods of time are independent•This means that variances are additive.•Standard deviations are not additive.

A WIENER PROCESS (1/3) A WIENER PROCESS (1/3) It is a particular type of Markov stochastic

process with a mean change of zero and a variance rate of 1.0 per year.

A variable z follows a Wiener Process if it has the following two properties:

(Property 1.) The change Δz during a small period of time

Δt is

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Wiener Process Wiener Process

(0,1) is where tzΔz~normal distribution

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A A WIENERWIENER PROCESS (2/3) PROCESS (2/3) (Property 2.)

The values of Δz for any two different short intervals of time, Δt, are independent.

Mean of z is 0

Variance of z is t

Standard deviation of z is

t

A WIENER PROCESS (3/3)A WIENER PROCESS (3/3)

Consider the change in the value of z during a relatively long period of time, T. This can be denoted by z(T)–z(0).

It can be regarded as the sum of the changes in z in N small time intervals of length t, where

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n

ii tzTz

1

)0()(

t

TN

Mean of [z (T ) – z (0)] is 0

Variance of [z (T ) – z (0)] is T

Standard deviation of [z (T ) – z (0)] is T

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EXAMPLE12.1(WIENEXAMPLE12.1(WIENER PROCESS) ER PROCESS) Ex ： Initially $25 and time is measured in years.

Mean ： 25 ， Standard deviation :1. At the end of 5 years, what is mean and Standard deviation?

Our uncertainty about the value of the variable at a certain time in the future, as measured by its standard deviation, increases as the square root of how far we are looking ahead.

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GENERALIZED GENERALIZED WIENER WIENER PROCESSES(1/3)PROCESSES(1/3)

A Wiener process, dz, that has been developed so far has a drift rate (i.e. average change per unit time) of 0 and a variance rate of 1

DR=0 means that the expected value of z at any future time is equal to its current value.

VR=1 means that the variance of the change in z in a time interval of length T equals T.

Drift rate →DR , variance rate →VR

DR=0 , VR=1

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GENERALIZED GENERALIZED WIENER PROCESSES WIENER PROCESSES (2/3)(2/3)

A generalized Wiener process for a variable x can be defined in terms of dz as

dx = a dt + b dz DR VR

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In a small time interval Δt, the change Δx in the value of x is given by equations

GENERALIZED GENERALIZED WIENER WIENER PROCESSES(3/3)PROCESSES(3/3)

tbtax

Mean of Δx is

Variance of Δx is

Standard deviation of Δx is tb

tb 2

ta

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EXAMPLE 12.2 EXAMPLE 12.2 Follow a generalized Wiener process

1. DR=20 (year) VR=900(year)

2. Initially , the cash position is 50.

3. At the end of 1 year the cash position will have a normal distribution with a mean of ★★ and standard deviation of ●●

ANS:★★=70, ●●=30

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ITÔ ITÔ PROCESS PROCESS

Itô Process is a generalized Wiener Itô Process is a generalized Wiener process in which the parameters a process in which the parameters a and b are functions of the value of and b are functions of the value of the underlying variable x and time t.the underlying variable x and time t.

dx=a(x,t) dt+b(x,t) dz

The discrete time equivalent

is only true in the limit as t tends to zero ttxbttxax ),(),(

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12.3 THE PROCESS 12.3 THE PROCESS FOR STOCKSFOR STOCKS

The assumption of constant expected drift rate is inappropriate and needs to be replaced by assumption that the expected reture is constant.

This means that in a short interval of time,Δt, the expected increase in S is μSΔt.

A stock price does exhibit volatility.

where is the expected return and is the volatility.

The discrete time equivalent is

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dzSdtSdS

tStSS

ANAN ITO PROCESS FOR STOCK ITO PROCESS FOR STOCK PRICESPRICES

dzdtS

dS

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EXAMPLE EXAMPLE 12.3 12.3

Suppose = 0.15, = 0.30, then

Consider a time interval of 1 week(0.0192)year, so that t =0.0192

ΔS=0.00288 S + 0.0416 S

ttS

S

dzdtS

dS

3.015.0

3.015.0

dzSdtSdS

dzdtS

dS

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MONTE CARLO MONTE CARLO SIMULATIONSIMULATION

MCS of a stochastic process is a procedure for sampling random outcome for the process.

Suppose = 0.14, = 0.2, and t = 0.01 then

The first time period(S=20 =0.52 ): S=0.0014*20 +0.02*20*0.52=0.236

The second time period ： S'=0.0014*20.236 +0.02*20.236*1.44=0.611

SSS 02.00014.0

tStSS

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MONTE CARLO MONTE CARLO SIMULATION – ONE SIMULATION – ONE PATHPATH

Week

Stock Price at Start of Period

Random Sample for

Change in Stock Price, S

0 20.00 0.52 0.236

1 20.236 1.44 0.611

2 20.847 -0.86 -0.329

3 20.518 1.46 0.628

4 21.146 -0.69 -0.262

tStSS

12.4 THE PARAMETERS12.4 THE PARAMETERS μ 、 σ

We do not have to concern ourselves with the determinants of μin any detail because the value of a derivative dependent on a stock is, in general, independent of μ.

We will discuss procedures for estimating σ in Chaper 13

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12.5 ITÔ12.5 ITÔ''S S LEMMALEMMA

If we know the stochastic process followed by x, Itô's lemma tells us the stochastic process followed by some function G (x, t )

dx=a(x,t)dt+b(x,t)dz

Itô's lemma shows that a functions G of x and t follows the process bdz

X

Gdtb

X

G

t

Ga

X

GdG

)2

1( 2

2

2

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DERIVATION OF DERIVATION OF ITÔITÔ''S LEMMA(1/2)S LEMMA(1/2)

Ifx is a small change in x and G is the resulting small change in G

...2

1

2

1

...6

1

2

1

22

222

2

2

33

32

2

2

yy

Gyx

yx

Gx

x

Gy

y

Gx

x

GG

yy

Gx

x

GG

xdx

Gdx

dx

Gdx

dX

dGG

XdX

dGG

dyy

Gdx

x

GdG

Taylor series

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DERIVATION OF ITÔ'S DERIVATION OF ITÔ'S LEMMA(2/2)LEMMA(2/2)

A Taylor's series expansion of G (x, t) gives

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22

22

2

½

½

tt

Gtx

tx

G

xx

Gt

t

Gx

x

GG

dztxbdttxadx ),(),(

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IGNORING TERMS OF IGNORING TERMS OF HIGHER ORDER THAN HIGHER ORDER THAN T T

t

x

xx

Gt

t

Gx

x

GG

tt

Gx

x

GG

½

22

2

order of

is whichcomponent a has because

becomes this calculus stochastic In

have wecalculusordinary In

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SUBSTITUTING SUBSTITUTING FOR ΔXFOR ΔX

tbx

Gt

t

Gx

x

GG

t

tbtax

dztxbdttxadx

222

2

½

n order thahigher of termsignoringThen

+ =

thatso

),(),(

Suppose

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THE ETHE E22ΔΔT T TERM TERM

tbx

Gt

t

Gx

x

GG

tt

ttE

E

EE

E

22

2

2

2

2

22

2

1

Hence ignored. be

can and toalproportion is of varianceThe

)( that followsIt

1)(

1)]([)(

0)(,)1,0( Since

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APPLICATION OF APPLICATION OF ITOITO''S LEMMA TO A S LEMMA TO A STOCK PRICE STOCK PRICE PROCESSPROCESS

dzSS

GdtS

S

G

t

GS

S

GdG

tSG

zdSdtSSd

½

and of function aFor

is process pricestock The

222

2

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APPLICATION TO APPLICATION TO FORWARD FORWARD CONTRACTS CONTRACTS

FdzFdtrdF

SdzedtrSeSedF

rSet

FS

F

eS

F

SeF

eSF

tTrtTrtTr

tTr

tTr

tTr

rT

)(

0

)()()(

)(

2

2

)(

)(

00

dzSS

GdtS

S

G

t

GS

S

GdG ½ 22

2

2

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THE LOGNORMAL THE LOGNORMAL PROPERTYPROPERTY

We define ：

dzdtdG

t

G

SS

G

SS

G

SG

2

0,1

,1

ln

2

22

2

dzSS

GdtS

S

G

t

GS

S

GdG ½ 22

2

2

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THE LOGNORMAL THE LOGNORMAL PROPERTYPROPERTY

The standard deviation of the logarithm of the stock price is

T

TTSS

TTSS

dzdtdG

T

T

22

0

22

0

2

,)2

(ln~ln

,)2

(~lnln

2