Model of the Behavior of Stock Prices Chapter 10
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Transcript of Model of the Behavior of Stock Prices Chapter 10
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10.1
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
Model of the Behavior
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 stock prices
• The continuous time, continuous variable process proves to be the most useful for the purposes of valuing derivative 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 future movements in a variable depend only on where we are, not the history of how we got where we are
• We will assume that stock prices follow 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 returns with a trading rule based on the past history of stock prices. In other words technical analysis does not work.
• A Markov process for stock prices is clearly consistent with weak-form market efficiency
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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 $40
• At the end of 1 year it is considered that it will have a probability distribution of(40,10) where (,) is a normal distribution with mean and standard deviation
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10.7
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
Questions• What is the probability distribution of the
stock price at the end of 2 years?
• ½ years?
• ¼ years?
• t years?
Taking limits we have defined a continuous variable, continuous time 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 in successive periods of time are independent
• 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 say that the variance is 100 per year.
• It is strictly speaking not correct to say 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
A Wiener Process (See pages 220-1)
• We consider a variable z whose value changes 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 (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 dt mean?
• It should be interpreted as meaning that the corresponding expression involving z and t is true in the limit as t tends to zero
• In this respect, stochastic calculus is analogous to ordinary calculus
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10.13
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
Generalized Wiener 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 process the drift rate & the variance rate can be set equal to any chosen constants
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10.14
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
Generalized Wiener Processes(continued)
The variable x follows a generalized Wiener process with a drift rate of a & a variance rate of b2 if
dx=adt+bdz
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10.15
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
Generalized Wiener Processes(continued)
• Mean change in x in time T is aT
• Variance of change in x in time T is b2T
• Standard deviation of change in x in time T is
x a t b t
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 probability distribution of(40,10) at the end of the year
• If we assume the stochastic process is Markov with 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-end distribution is (48,10), the process is
dS = 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 the variance 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 t tends to
zero
x a x t t b x t t ( , ) ( , )
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10.18
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
Why a Generalized Wiener Processis not Appropriate for Stocks
• For a stock price we can conjecture that its expected proportional change in a short period of time remains constant not its expected absolute change in a short period of time
• We can also conjecture that our uncertainty as to the size of future stock price movements is proportional 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 is the expected return is the volatility.
The discrete time equivalent is
dS Sdt Sdz
S S t S t
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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 for
• Suppose = 0.14, = 0.20, and t = 0.01, then
S S S 0 0014 0 02. .
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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)
PeriodStock Price atStart of Period
RandomSample for
Change in StockPrice, S
0 20.000 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
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10.22
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
Ito’s Lemma (See pages 229-231)
• If we know the stochastic process followed by x, Ito’s lemma tells us the stochastic process followed by some function G (x, t )
• Since a derivative security is a function of the price of the underlying & time, Ito’s lemma plays an important part in the analysis of derivative securities
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10.23
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
Taylor Series Expansion• A Taylor’s series expansion of G (x , t ) gives
GG
xx
G
tt
G
xx
G
x tx t
G
tt
½
½
2
22
2 2
22
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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 ordinary calculus we get
stic calculus we get
because has a component which is of order
In stocha
½
GG
xx
G
tt
GG
xx
G
tt
G
xx
x t
2
22
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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
½
dx a x t dt b x t dz
x a t b t
t
GG
xx
G
tt
G
xb t
2
22 2
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10.26
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
The 2t Term
Since
It follows that
The variance of is proportional to and can
be ignored. Hence
2
( , ) ( )
( ) [ ( )]
( )
( )
0 1 0
1
1
1
2
2 2
2
2
2
22
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 Limits
Taking limits ½
Substituting
We obtain ½
This is Ito's Lemma
dGG
xdx
G
tdt
G
xb dt
dx a dt b dz
dGG
xa
G
t
G
xb dt
G
xb dz
2
22
2
22
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10.28
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
Application of Ito’s Lemmato a Stock Price Process
The stock price process is
For a function of &
½
d S S dt S d z
G S t
dGG
SS
G
t
G
SS dt
G
SS dz
2
22 2
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10.29
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
Examples1. The forward price of a stock for a contract
maturing at time
e
2.
T
G S
dG r G dt G dz
G S
dG dt dz
r T t
( )
( )
ln
2
2