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10.1
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Model of theBehavior
of Stock Prices
Chapter 10
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10.2
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Categorization of Stochastic
Processes
Discrete time; discrete variable
Discrete time; continuous variable Continuous time; discrete variable Continuous time; continuous variable
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10.3
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Modeling Stock Prices We can use any of the four types of
stochastic processes to model stockprices
The continuous time, continuousvariable process proves to be the most
useful for the purposes of valuingderivative securities
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10.4
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Markov Processes (See pages 218-9)
In a Markov process futuremovements in a variable depend onlyon where we are, not the history ofhow we got where we are
We will assume that stock pricesfollow Markov processes
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10.5
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Weak-Form Market
Efficiency The assertion is that it is impossible to
produce consistently superior returnswith a trading rule based on the past
history of stock prices. In other wordstechnical analysis does not work.
A Markov process for stock prices isclearly consistent with weak-form marketefficiency
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10.6
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Example of a Discrete Time
Continuous Variable Model
A stock price is currently at $40At the end of 1 year it is
considered that it will have aprobability distribution of
J(40,10) where J(Q,W) is anormal distribution with mean Qand standard deviation W
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10.7
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Questions
What is the probability distribution ofthe stock price at the end of 2years?
years? years? (tyears?Taking limits we have defined acontinuous variable, continuoustime process
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10.8
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
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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10.9
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Variances & Standard
Deviations (continued)
In our example it is correct to saythat the variance is 100 per year.
It is strictly speaking not correct tosay that the standard deviation is 10
per year.
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10.10
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
AWiener Process (See pages 220-1)
We consider a variable z whose valuechanges continuously
The change in a small interval of time (t is(z
The variable follows a Wiener process if
1.2. The values of(z for any 2 different (non-overlapping) periods of time are independent
( (z t! I I J(0,1)where is a random drawing from
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10.11
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
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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10.12
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Taking Limits ...
What does an expression involving dz and dtmean?
It should be interpreted as meaning that thecorresponding expression involving (z and (tis
true in the limit as (t tends to zero
In this respect, stochastic calculus is analogous toordinary calculus
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10.13
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
GeneralizedWiener Processes(See page 221-4)
A Wiener process has a drift rate(ie average change per unit time) of
0 and a variance rate of 1
In a generalized Wiener processthe drift rate & the variance rate
can be set equal to any chosenconstants
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10.14
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
GeneralizedWiener Processes(continued)
The variable x follows a generalized
Wiener process with a drift rate ofa& a variance rate ofb2 if
dx=adt+bdz
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10.15
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
GeneralizedWiener Processes(continued)
Mean change in x in time T isaT
Variance of change in x in time Tis b2T
Standard deviation of change inx in time T is
( ( (x a t b t ! I
b T
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10.16
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
The Example Revisited
A stock price starts at 40 & has a probabilitydistribution ofJ(40,10) at the end of the year
If we assume the stochastic process is Markov
with no drift then the process isdS = 10dz
If the stock price were expected to grow by $8on average during the year, so that the year-
end distribution is J(48,10), the process isdS = 8dt + 10dz
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10.17
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Ito Process (See pages 224-5)
In an Ito 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 (ttends to
zero
( ( ( x a x t t b x t t! ( , ) ( , )I
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10.18
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Why a GeneralizedWiener Process
is not Appropriate for Stocks
For a stock price we can conjecture that itsexpected proportional change in a short period
of time remains constant not its expectedabsolute change in a short period of time
We can also conjecture that our uncertainty asto the size of future stock price movements isproportional to the level of the stock price
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10.19
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
An Ito Process for Stock Prices(See pages 225-6)
where Q is the expected return Wis the volatility.
The discrete time equivalent is
dS Sdt Sdz! Q W
( ( (S S t S t ! Q W I
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10.20
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Monte Carlo Simulation We can sample random paths for the
stock price by sampling values forI
Suppose Q= 0.14, W= 0.20, and (t=0.01, then
(S S S! 0 0014 0 02. . I
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10.21
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Monte Carlo Simulation One Path(continued. See Table 10.1)
Perit ck Price tt rt f Peri
RleforI
Change in t ckPrice, (
. .5 . 36
. 36 . .6
. 7 - . 6 - .3 9
3 .5 . 6 .6
. 6 - .69 - . 6
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10.22
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Itos Lemma (See pages 229-231) If we know the stochastic process
followed by x, Itos lemma tells us the
stochastic process followed by somefunction G (x, t)
Since a derivative security is a function ofthe price of the underlying & time, Itos
lemma plays an important part in theanalysis of derivative securities
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10.23
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Taylor Series ExpansionA Taylors series expansion of
G (x , t) gives
( ( ( (
( ( (
GG
xx
G
tt
G
xx
G
x tx t
G
tt
!
x
x
x
x
x
x
x
x x
x
x
2
2
2
2 2
2
2-
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10.24
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Ignoring Terms of Higher Order
Than (t
In r inar calc l s eget
stic calc l s eget
ecause has a c nent hich is f r er
In st cha
( ( (
( ( ( (
( (
G
G
xx
G
t t
G
G
x x
G
t t
G
x x
x t
!
!
x
x
x
x
x
x
x
x
x
x
2
2
2
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10.25
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Substituting for (x
Suppose
( , ) ( , )
so that= +
Then ignoring terms of higher order than
dx a x t dt b x t dz
x a t b t
t
GG
xx
G
tt
G
xb t
!
!
( ( (
(
( ( ( (
I
x
x
x
x
x
xI
2
2
2 2
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10.26
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
The I2(tTerm
Since
It follows that
The variance of is proportional to and can
be ignored. Hence
2
I J II I
I
I
x
x
x
x
x
x
} ! !
!
!
!
( , ) ( )
( ) [ ( )]
( )
( )
0 1 0
1
1
1
2
2 2
2
2
2
2
2
E
E E
E
E t t
t t
GG
xx
G
tt
G
xb t
( (
( (
( ( ( (
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10.27
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Taking LimitsTaking limits
Substituting
We obtain
This is Ito's Lemma
dGG
xdx
G
tdt
G
xb dt
dx a dt b d z
dGG
xa
G
t
G
xb dt
G
xb dz
!
!
!
x
x
x
x
x
x
x
x
x
x
x
x
x
x
2
2
2
2
2
2
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10.28
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Application ofItos Lemma
to a Stock Price ProcessThe stock price process is
For a function of &
d S S dt S d z
G S t
dGG
SS
G
t
G
SS dt
G
SS dz
!
!
Q W
xx
Qxx
x
xW
xx
W2
2
2 2
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10.29
Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull
Examples
1. The forward price of a stock for a contractmaturing at time
e
2.
T
G S
dG rG dt G dz
G S
dG dt dz
r T t!
!
!
!
( )
( )
ln
Q W
Q W W2
2