Ch12HullOFOD7thEd.ppt

8/14/2019 Ch12HullOFOD7thEd.ppt

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Wiener Processes and Its

LemmaChapter 12

1

Options, Futures, and Other

Derivatives, 7th Edition, Copyright John C. Hull 2008


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Derivatives, 7thEdition, Copyright John C. Hull 2008 2

Types of Stochastic Processes

Discrete time; discrete variable

Discrete time; continuous variable

Continuous time; discrete variable

Continuous time; continuous variable


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Modeling Stock Prices

We can use any of the four types ofstochastic processes to model stockprices

The continuous time, continuous variableprocess proves to be the most useful forthe purposes of valuing derivatives


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Markov Processes (See pages 259-60)

In a Markov process future movementsin a variable depend only on where weare, not the history of how we got

where we are We assume that stock prices follow

Markov processes


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Weak-Form Market Efficiency

This asserts that it is impossible toproduce consistently superior returns witha trading rule based on the past history of

stock prices. In other words technicalanalysis does not work.

A Markov process for stock prices is

consistent with weak-form marketefficiency


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Example of a Discrete TimeContinuous Variable Model

A stock price is currently at $40

At the end of 1 year it is considered that itwill have a normal probability distribution ofwith mean $40 and standard deviation $10


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Questions

What is the probability distribution ofthe stock price at the end of 2 years?

years?

years?

Dt years?

Taking limits we have defined a

continuous variable, continuous timeprocess


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Variances & Standard Deviations

In Markov processes changes insuccessive periods of time areindependent

This means that variances are additive

Standard deviations are not additive


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Variances & Standard Deviations(continued)

In our example it is correct to say that thevariance is 100 per year.

It is strictly speaking not correct to saythat the standard deviation is 10 per year.


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A Wiener Process (See pages 261-63)

We consider a variablez whose valuechanges continuously

Definef(m,v)as a normal distribution with

mean mand variance v The change in a small interval of time Dt is Dz

The variable follows a Wiener process if

The values of Dz for any 2 different (non-overlapping) periods of time are independent

(0,1)iswhere fDD tz


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Properties of a Wiener Process

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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Taking Limits . . .

What does an expression involving dz anddt mean?

It should be interpreted as meaning that thecorresponding expression involving Dz andDtis true in the limit as Dt tends to zero

In this respect, stochastic calculus is

analogous to ordinary calculus


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Generalized Wiener Processes(See page 263-65)

A Wiener process has a drift rate (i.e.average change per unit time) of 0 and avariance rate of 1

In a generalized Wiener process the driftrate and the variance rate can be set equalto any chosen constants


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Generalized Wiener Processes(continued)

The variablexfollows a generalized Wienerprocess with a drift rate of aand a variance

rate ofb2ifdx=a dt+b dz


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Generalized Wiener Processes(continued)

Mean change inx in time Tis aT

Variance of change inxin timeTis b2T

Standard deviation of change inxintime Tis

tbtax DDD

b T


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The Example Revisited

A stock price starts at 40 and has a probabilitydistribution off(40,100) at the end of the year

If we assume the stochastic process is Markovwith no drift then the process is

dS = 10dz If the stock price were expected to grow by $8 on

average during the year, so that the year-enddistribution is f(48,100), the process would be

dS = 8dt + 10dz


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It Process (See pages 265)

In an It process the drift rate and thevariance rate are functions of time

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

The discrete time equivalent

is only true in the limit as Dttends to

zero

ttxbttxax DDD ),(),(


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Why a Generalized WienerProcess Is Not Appropriate forStocks

For a stock price we can conjecture that itsexpected percentage change in a shortperiod of time remains constant, not itsexpected absolute change in a shortperiod of time

We can also conjecture that ouruncertainty as to the size of future stockprice movements is proportional to thelevel of the stock price


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An Ito Process for Stock Prices(See pages 269-71)

where mis the expected return sis thevolatility.

The discrete time equivalent is

dzSdtSdS sm

tStSS DsDmD


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Monte Carlo Simulation

We can sample random paths for the stockprice by sampling values for

Suppose m= 0.15, s= 0.30, and Dt= 1

week (=1/52 years), then

SSS 0416.000288.0 D


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Monte Carlo SimulationOne Path (SeeTable 12.1, page 268)

WeekStock Price atStart of Period

Random

Sample for

Change in Stock

Price, DS

0 100.00 0.52 2.45

1 102.45 1.44 6.43

2 108.88 -0.86 -3.58

3 105.30 1.46 6.70

4 112.00 -0.69 -2.89


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ItsLemma (See pages 269-270)

If we know the stochastic processfollowed byx, Its lemma tells us thestochastic process followed by somefunction G(x, t)

Since a derivative is a function of theprice of the underlying and time, Its

lemma plays an important part in theanalysis of derivative securities


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Taylor Series Expansion

A Taylors series expansion of G(x, t)gives

D

DD

D

D

D

D

2

2

22

2

2

2

t

t

Gtx

tx

G

xx

Gt

t

Gx

x

GG


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Ignoring Terms of Higher Order

Than Dt

t

x

xx

Gt

t

Gx

x

GG

t

t

Gx

x

GG

DD

D

D

D

D

D

D

D

orderofiswhichcomponentahasbecause

becomesthiscalculusstochasticIn

havewecalculusordinaryIn

2

2

2


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Substituting for Dx

tbx

Gt

t

Gx

x

GG

t

tbtax

dztxbdttxadx

D

D

D

D

DDDD

22

2

2

thanorderhigheroftermsignoringThen+=

thatso

),(),(

Suppose


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The 2Dt Term

tbx

G

tt

G

xx

G

G

ttttE

E

EE

E

D

D

D

D

DDDD

f

2

2

2

2

2

22

2

1

)(

1)(

1)]([)(

0)(,)1,0(

Henceignored.be

canandtoalproportionisofvarianceThethatfollowsIt

Since

2


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Taking Limits

LemmasIto'isThis

:obtainWe

:ngSubstituti

:limitsTaking

dzbx

Gdtbx

G

t

Gax

GdG

dzbdtadx

dtbx

Gdt

t

Gdx

x

GdG

2

2

2

2

2

2


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Application of Itos Lemmato a Stock Price Process

dzSS

GdtS

S

G

t

GS

S

GdG

tSG

zdSdtSSd

andoffunctionaFor

isprocesspricestockThe

s

s

m

sm

22

2

2


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Derivatives, 7thEdition, Copyright J h C H ll 2008 29

Examples

dzdtdG

SG

dzGdtGrdGeSG

TtTr

2.

timeatmaturing

contractaforstockaofpriceforwardThe1.

s

s

m

sm

2

ln

)(

2

)(

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