wiener process and ito's lemma.ppt

WIENER PROCESSES
AND ITÔ’S LEMMA
OPTIONS, FUTURES, AND OTHER DERIVATIVES, 8TH EDITION,
COPYRIGHT © JOHN C. HULL 2012
1

STOCHASTIC PROCESSES
Describes the way in which a variable such as a stock price,
exchange rate or interest rate changes through time
Incorporates uncertainties
2

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%
3

EXAMPLE 2
Each day a stock price change is drawn from a normal
distribution with mean $0.2 and standard deviation $1
4

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
We assume that stock prices follow Markov processes
5

A stochastic process { Xn } is called a Markov chain if
Pr{Xn+1 = j | X0 = k0, . . . , Xn-1 = kn-1, Xn = i }
= Pr{ Xn+1 = j | Xn = i }  transition probabilities
for every i, j, k0, . . . , kn-1 and for every n.
The future behavior of the system depends only on the
current state i and not on any of the previous states.
MARKOV CHAIN DEFINITION

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
7

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
8

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
9

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
10

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.
11

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
The values of Dz for any 2 different (non-overlapping) periods of time are
independent
12
(0,1)
is
where f

D


D t
z

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
13
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
14

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
15

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
16
z
b
t
a
x
t
b
t
a
x
D

D

D
D

D

D 
b T

OPTIONS, FUTURES, AND OTHER DERIVATIVES, 8TH EDITION, COPYRIGHT © JOHN
C. HULL 2012
17

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
18

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
is true in the limit as Dt tends to
zero
19
t
t
x
b
t
t
x
a
x D


D

D )
,
(
)
,
(

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
20

AN ITO PROCESS FOR STOCK
PRICES
(SEE PAGES 286-89)
where m is the expected return s is the volatility.
The discrete time equivalent is
The process is known as geometric Brownian motion
21
dz
S
dt
S
dS s

m

t
S
t
S
S D

s

D
m

D

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
22

TAYLOR SERIES EXPANSION
A Taylor’s series expansion of G(x, t) gives
24


D



D
D




D



D



D



D
2
2
2
2
2
2
2
t
t
G
t
x
t
x
G
x
x
G
t
t
G
x
x
G
G
½
½

IGNORING TERMS OF HIGHER
ORDER THAN DT
25
t
x
x
x
G
t
t
G
x
x
G
G
t
t
G
x
x
G
G
D
D
D



D



D



D
D



D



D
order
of
is
which
component
a
has
because
½
becomes
this
calculus
stochastic
In
have
we
calculus
ordinary
In
2
2
2

TAKING LIMITS
26
Lemma
s
Ito'
is
This
½
:
obtain
We
:
ng
Substituti
½
:
limits
Taking
2
2
2
2
2
2
dz
b
x
G
dt
b
x
G
t
G
a
x
G
dG
dz
b
dt
a
dx
dt
b
x
G
dt
t
G
dx
x
G
dG
































APPLICATION OF ITO’S
LEMMA
TO A STOCK PRICE PROCESS
27
dz
S
S
G
dt
S
S
G
t
G
S
S
G
dG
t
S
G
z
d
S
dt
S
S
d
½
and
of
function
a
For
is
process
price
stock
The
s











s






m



s

m

2
2
2
2

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