Transcript A Wiener Process

Chapter 13
Wiener Processes and Itô’s
Lemma
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Stochastic Processes
Describes the way in which a variable such
as a stock price, exchange rate or interest
rate changes through time
Incorporates uncertainties
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Example 1
Each day a stock price
increases by $1 with probability 30%
stays the same with probability 50%
reduces by $1 with probability 20%
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Example 2
Each day a stock price change is drawn from
a normal distribution with mean $0.2 and
standard deviation $1
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Markov Processes (See pages 280-81)
In a Markov process future movements in a
variable depend only on where we are, not
the history of how we got to where we are
Is the process followed by the temperature
at a certain place Markov?
We assume that stock prices follow Markov
processes
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Weak-Form Market Efficiency
This asserts 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
consistent with weak-form market
efficiency
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Example
A variable is currently 40
It follows a Markov process
Process is stationary (i.e. the parameters of
the process do not change as we move
through time)
At the end of 1 year the variable will have a
normal probability distribution with mean 40
and standard deviation 10
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Questions
What is the probability distribution of
the stock price at the end of 2 years?
½ years?
¼ years?
Dt years?
Taking limits we have defined a
continuous stochastic process
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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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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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A Wiener Process (See pages 282-84)
Define f(m,v) as a normal distribution with
mean m and variance v
A variable z follows a Wiener process if
The change in z in a small interval of time Dt
is Dz
Dz   Dt where  is f(0,1)
The values of Dz for any 2 different (nonoverlapping) periods of time are independent
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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
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Generalized Wiener Processes
(See page 284-86)
A Wiener process has a drift rate (i.e.
average change per unit time) of 0 and a
variance rate of 1
In a generalized Wiener process the drift
rate and the variance rate can be set equal
to any chosen constants
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Generalized Wiener Processes
(continued)
Dx  a Dt  b  Dt
Mean change in x per unit time is a
Variance of change in x per unit time is
b2
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Taking Limits . . .
What does an expression involving dz and dt
mean?
It should be interpreted as meaning that the
corresponding expression involving Dz and Dt
is true in the limit as Dt tends to zero
In this respect, stochastic calculus is
analogous to ordinary calculus
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The Example Revisited
A stock price starts at 40 and has a probability
distribution of f(40,100) 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 f(48,100), the process would be
dS = 8dt + 10dz
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Itô Process (See pages 286)
In an Itô 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
Dx  a ( x, t )Dt  b( x, t ) Dt
is true in the limit as Dt tends to
zero
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Why a Generalized Wiener Process
Is Not Appropriate for Stocks
For a stock price we can conjecture that its
expected percentage change in a short period
of time remains constant (not its expected
actual change)
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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An Ito Process for Stock Prices
(See pages 286-89)
dS  mS dt  sS dz
where m is the expected return s is the
volatility.
The discrete time equivalent is
DS  mSDt  sS Dt
The process is known as geometric
Brownian motion
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Interest Rates
What would be a reasonable stochastic
process to assume for the short-term interest
rate?
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Monte Carlo Simulation
We can sample random paths for the stock
price by sampling values for 
Suppose m= 0.15, s= 0.30, and Dt = 1 week
(=1/52 or 0.192 years), then
ΔS  0.15  0.0192 S  0.30  0.0192 ε
or
DS  0.00288 S  0.0416 S
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Monte Carlo Simulation – Sampling one
Path (See Table 13.1, page 289)
Week
Stock Price at
Random
Start of Period 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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Correlated Processes
Suppose dz1 and dz2 are Wiener processes with
correlation r
Then
Dz1  1 Dt
Dz 2   2 Dt
where 1 and  2 are random samples
from a bivariate standard normal
distribution where correlation is r
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Itô’s Lemma (See pages 291)
If we know the stochastic process
followed by x, Itô’s lemma tells us the
stochastic process followed by some
function G (x, t )
Since a derivative is a function of the
price of the underlying asset and time,
Itô’s lemma plays an important part in
the analysis of derivatives
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Taylor Series Expansion
A Taylor’s series expansion of G(x, t)
gives
G
G
 2G
DG 
Dx 
Dt  ½ 2 Dx 2
x
t
x
 2G
 2G 2

Dx Dt  ½ 2 Dt  
xt
t
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Ignoring Terms of Higher Order
Than Dt
In ordinary calculus we have
G
G
DG 
Dx 
Dt
x
t
In stochastic calculus this becomes
G
G
 2G 2
DG 
Dx 
Dt  ½
Dx
2
x
t
x
because Dx has a component which is
of order Dt
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Substituting for Dx
Suppose
dx  a( x, t )dt  b( x, t )dz
so that
Dx = a Dt + b  Dt
Then ignoring terms of higher order than Dt
G
G
 2G 2 2
DG 
Dx 
Dt  ½
b  Dt
2
x
t
x
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The 2Dt Term
Since   f(0,1), E ()  0
E ( 2 )  [ E ()]2  1
E ( 2 )  1
It follows that E ( 2 Dt )  Dt
The variance of Dt is proportion al to Dt 2 and can
be ignored. Hence
G
G
1  2G 2
DG 
Dx 
Dt 
b Dt
2
x
t
2 x
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Taking Limits
Taking limits :
Substituti ng :
We obtain :
G
G
 2G 2
dG 
dx 
dt  ½ 2 b dt
x
t
x
dx  a dt  b dz
 G
G
 2G 2 
G
dG  
a
 ½ 2 b dt 
b dz
t
x
x
 x

This is Ito' s Lemma
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Application of Ito’s Lemma
to a Stock Price Process
The stock price process is
d S  mS dt  sS d z
For a function G of S and t
 G
G
 2G 2 2 
G
dG  
mS 
 ½ 2 s S dt 
sS dz
t
S
S
 S

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Examples
1. The forward price of a stock for a contract
maturing at time T
G  S e r (T  t )
dG  (m  r )G dt  sG dz
2. The log of a stock price
G  ln S

s2 
dt  s dz
dG   m 

2


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