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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Examples
dzdtdG
SG
dzGdtGrdGeSG
TtTr
2.
timeatmaturing
contractaforstockaofpriceforwardThe1.
s
s
m
sm
2
ln
)(
2
)(