Wiener Processes and Itô’s
Lemma
1
A Wiener Process
We consider a variable z whose value
changes continuously
Define f(m,v) as a normal distribution with
mean m and variance v
The change in a small interval of time Dt is Dz
The variable follows a Wiener process if






Dz   Dt where  is f(0,1)
The values of Dz for any 2 different (nonoverlapping) periods of time are independent
2
Properties of a Wiener Process
Mean of Δz=0
 Standard deviation of Δz=
Dt
 Variance of Δz= Δt
z follows a Markov 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
3
Wiener process
4
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
5
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
b  Dt term add noise or variability
to the path followed by x.
6
Generalized Winner Process
7
Generalized Wiener Processes
(continued)



Mean of [x (T ) – x (0)] is aT
Variance of [x (T ) – x (0)] is b2T
Standard deviation of [x(T ) – x (0)]
b*sqrt(T)
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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
Generalized Winner Process:
dx=adt+bdz
9
Generalized Wiener Processes



Mean of [x (T ) – x (0)] is aT
Variance of [x (T ) – x (0)] is b2T
Standard deviation of [x(T ) – x (0)] b*sqrt(T)
10
Itô Process


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 only true in the limit as Dt tends to
zero
Ito process is also Markov.
11
An Ito Process for Stock Prices
(See pages 286-89)
It is reasonable to assume that the percentage
return in a short period of time is random, but
its variability is the same regardless of the
stock price. Thus we have
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
12
Monte Carlo Simulation

Fro the process of stock price, we have
DS
S


f ( mDt , s 2 Dt )
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 years), then
DS  0.00288S  0.0416S
13
Monte Carlo Simulation – One Path
Excel code : NORMINV(RAND(),0,1)
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
14
Monte Carlo Simulation


Monte Carlo methods use statistics gathered from random sampling
to model and simulate a complicated distribution.
“If “the future” is the random selection of one sequence of events
from a probability space, our focus should revolve around modeling
the distribution more so than identifying the ultimate future path.”
15
Itô’s Lemma


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 and time, Itô’s
lemma plays an important part in the
analysis of derivative securities
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Itô’s Lemma-formal argument
DG 
17
Itô’s Lemma-formal argument
18
Itô’s Lemma-formal argument
Since    (0,1) ,
E ( )  0
E ( 2 )  [ E ( )]2  1
E ( 2 )  1
It follows that E ( 2 Dt )  Dt
The variance of  2 Dt is proportional to Dt 2 and can
be ignored. Hence
G
G
1  2G 2
DG 
Dx 
Dt 
b Dt
2
x
t
2x
19
Application of Ito’s Lemma
to a Stock Price Process

We argued that under a reasonable assumption:
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


Both S and G are affected by dz - the noise in
the system.
20
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. G  ln S

s2 
dt  s dz
dG   m 
2 

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