Chapter
12
Wiener Processes
and Ito’s Lemma
Stochastic Processes
A
stochastic process describes the way a variable
evolves over time that is at least in part random. i.e.,
temperature and IBM stock price
 A stochastic process is defined by a probability law for
the evolution of a variable xt over time t. For given
times, we can calculate the probability that the
corresponding values x1,x2, x3,etc. lie in some specified
range.
Categorization of Stochastic
Processes
 Discrete
time; discrete variable
Random walk:
t
t 1
t
if  tcan only take on discrete values
 Discrete time; continuous variable
x  x 
xt  a  bxt 1   t
 t is a normally distributed random variable with
zero mean.
 Continuous time; discrete variable
 Continuous time; continuous variable
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
Markov Processes
 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.
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
Example of a Discrete Time Continuous
Variable Model
A
stock price follows a Markov process, and
is currently at $40.
 At the end of 1 year it is considered that it
will have a probability distribution of
f(40,10) where f(m,s) is a normal
distribution with mean m and standard
deviation s.
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 variable, continuous time
process
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
Variances & Standard Deviations
(continued)
 In
our example it is correct to say that the
variance is 100 per year.
 It is strictly not correct to say that the
standard deviation is 10 per year.
A Wiener Process (Brownian
Motion)
 We
consider a variable z whose value changes
continuously
 The change in a small interval of time Dt is Dz
 The variable follows a Wiener process if
1. Dz   Dt where  is a random drawing from f(0,1)
2. The values of Dz for any 2 different (nonoverlapping) periods of time are independent
Properties of a Wiener Process
of [z (T ) – z (0)] is 0
 Variance of [z (T ) – z (0)] is T
 Mean
z (T )  z (0)    i Dt
n
i 1
 Standard
deviation of [z (T ) – z (0)] is
T
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
dz   dt
Generalized Wiener Processes
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
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
or: x(t)=x0+at+bz(t)
Generalized Wiener Processes
(continued)
Dx  a Dt  b  Dt
 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
b
T
The Example Revisited
A
stock price starts at 40 & has a probability
distribution of f(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 f(48,10), the process is
dS = 8dt + 10dz
Whyb Dt ?(1)
 It’s
the only way to make the variance of (xTx0)depend on T and not on the number of steps.
1.Divide time up into n discrete periods of length
△ t, n=T/ △ t. In each period the variable x
either moves up or down by an amount △ h
with the probabilities of p and q respectively.
Why b
?(2)
Dt
2.the distribution for the future values of x:
E(△x)=(p-q) △ h
E[(△ x)2]= p(△ h)2+q(- △ h)2
So, the variance of △ x is:
E[(△ x)2]-[E(△ x)]2=[1-(p-q)2](△ h)2=4pq(△ h)2
3. Since the successive steps of the random walk are
independent, the cumulated change(xT-x0)is a binomial
random walk with mean:
n(p-q) △ h=T(p-q) △ h/ △ t
and variance: n [1-(p-q)2](△ h)2= 4pqT(△ h)2 / △ t
Whyb Dt ?(3)
let △ t go to zero, we would like the mean and
variance of (xT-x0) to remain unchanged, and to be
independent of the particular choice of p,q, △ h and △
t.
 The only way to get it is to set:
 When
and
then
Dh  b Dt
a
a
1
1
p  [1 
Dt ], q  [1 
Dt ]
2
2
b
b
a
a
pq 
Dt  2 Dh
b
b
Whyb Dt ?(4)
△ t goes to zero,the binomial distribution
converges to a normal distribution, with mean
 When
and variance
a
Dh
t 2 Dh
 at
b
Dt
a 2 b 2 Dt
t[1  ( ) Dt ]
 b 2t
b
Dt
Sample path(a=0.2 per year,b2=1.0 per
year)
 Taking
a time interval of one month, then
calculating a trajectory for xt using the equation:
xt  xt 1  0.01667  0.2887 t
A trend of 0.2 per year implies a trend of 0.0167
per month. A variance of 1.0 per year implies a
variance of 0.0833 per month, so that the
standard deviation in monthly terms is 0.2887.
See Investment under uncertainty, p66
Forecast using generalized Brownian
Motion
 Given
the value of x(t)for Dec. 1974,X1974 , the
forecasted value of x for a time T months
beyond Dec. 1974 is given by:
xˆ1974 T  x1974  0.01667T
See Investment under uncertainty, p67
 In the long run, the trend is the dominant
determinant of Brownian Motion, whereas in the
short run, the volatility of the process dominates.
Why a Generalized Wiener Processes
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
 The price of a stock never fall below zero.
Ito Process
 In
an Ito process the drift rate and the variance rate
are functions of time
dx=a(x,t)dt+b(x,t)dz
t
t
x(t )  x0  0 ads  0 bdz
or:
 The discrete time equivalent
Dx  a ( x , t ) Dt  b( x , t ) Dt
is only true in the limit as Dt tends to
zero
An Ito Process for Stock Prices
dS  mSdt  sSdz
where m is the expected return s is the
volatility.
The discrete time equivalent is
DS  mSDt  sS Dt
Monte Carlo Simulation
 We
can sample random paths for the stock
price by sampling values for 
 Suppose m= 0.14, s= 0.20, and Dt = 0.01, then
DS  0.0014 S  0.02 S
Monte Carlo Simulation – One Path
Stock Price at
Start of Period
Random
Sample for 
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
Period
Change in Stock
Price, DS
Ito’s Lemma
 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
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
Ignoring Terms of Higher Order Than Dt
In ordinary calculus we get
G
G
DG 
Dx
Dt
x
t
In stochastic calculus we get
G
G
 2G
2
DG 
D x
Dt ½
D
x
x
t
 x2
because Dx has a component which is of order Dt
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
 G 2 2
DG 
Dx 
Dt  ½ 2 b  Dt
x
t
x
2
The 2Dt Term
Since  ~ f (0,1) E ( )  0
E ( )  [ E ( )]  1
2
2
E ( 2 )  1
It follows that E ( 2 Dt )  Dt
The variance of Dt is proportion al to Dt and can
be ignored. Hence
2
G
G
1 G 2
DG 
Dx 
Dt 
b Dt
2
x
t
2 x
2
Taking Limits
Taking limits
G
G
2G 2
dG 
dx 
dt  ½ 2 b dt
x
t
x
Substituting
dx  a dt  b dz
We obtain
 G
G
 2G 2 
G
dG  
a
 ½ 2 b  dt 
b dz
t
x 
x
 x
This is Ito's Lemma
Application of Ito’s Lemma
to a Stock Price Process
The stock price process is
d S  m S dt  s S d z
For a function G of S & t
 G
G
 2G 2 2 
G
dG  
mS 
½
s S dz
2 s S  dt 
t
S
S
S

Examples
1. The forward price of a stock for a contract
maturing at time T
G  S er ( T  t )
dG  (m  r )G dt  sG dz
2. G  ln S

s2 
dG   m   dt  s dz
2

Ito’s Lemma for several Ito processes
F=F(x1,x2,…,xm,xt) is a function of time
and of the m Ito process x1,x2,…,xm,
where
dxi=ai(x1,x2,…,xm,t)dt+bi(x1,x2,…,xm,t)dzi,i=1,…,
m,with E(dzidzj)= ρijdt.Then Ito’s Lemma gives
the defferential dF as
 Suppose
2
F

F
1

F
dF 
dt  i dxi  i j
dxi dx j
t
xi
2 xi x j
Examples
Suppose F(x,y)=xy, where x and y each follow geometric
Brownian motions:
dx=axxdt+bxxdzx
dy=ayydt+byydzy
with E(dzxdzy)=ρdt. What’s the process followed by F(x,y) and
by G=logF?
dF=xdy+ydx+dxdy
=(ax+ay+ ρbxby)Fdt+(bxdzx+bydzy)F
dG= (ax+ay-1/2bx2-1/2by2)dt+bxdzx+bydzy