Download Wiener Processes and Ito`s Lemma

CH12WIENER PROCESSES AND ITÔ'S
LEMMA
OUTLINE
• The Markov Property
.*.
• Continuous-Time Stochastic Processes
• The Process For a Stock Price
• The Parameters
• Itô's Lemma
• The Lognormal Property
1
Continuous
Variable
the value of the variable changes
only at certain fixed time point
.*.
Stochastic
Process
Discrete
Variable
only limited values are possible for
the variable
2
12.1 THE MARKOV
PROPERTY

A Markov process is a particular type of
stochastic process .
where only the present value of a variable is
relevant for predicting the future.

The past history of the variable and the
way that the present has emerged from
the past are irrelevant.
4
CONTINUOUS-TIME
STOCHASTIC PROCESSES

Suppose
$10(now), change in its value
•In Markov processes changes in
during
1 yearN~(μ=0,
is f(0,1).
σ=1)
successive
periods
of time are
independent
•This means that variances are
Whatadditive.
is the probability distribution
of theσ=2)
N~(μ=0,
stock
price atdeviations
the endare
of not
2 years?
•Standard
additive.



f(0,2)
6 months? f(0,0.5)
3 months? f(0,0.25)
Dt years? f(0, Dt)
5
A WIENER PROCESS (1/3)

It is a particular type of Markov stochastic process
with a mean change of zero and a variance rate of
1.0 per year.

.*.
Wiener Process
A variable z follows a Wiener Process if it has the
following two properties:
(Property 1.)
The change Δz during a small period of time Δt is
Dz  Dt where  is f (0,1)
Δz~normal distribution
7
A WIENER PROCESS
(2/3)
(Property 2.)
The values of Δz for any two different
short intervals of time, Δt, are
independent.

Mean of Dz is 0
Dt

Variance of Dz is Dt

Standard deviation of Dz is
7
A WIENER PROCESS (3/3)
Mean of [z (T ) – z (0)] is 0
 Consider the change in the value of z during a
relatively
time,
Variancelong
of [zperiod
(T ) – of
z (0)]
isT.
T This can be
denoted by z(T)–z(0).
deviation
of [zsum
(T ) –ofzthe
(0)]changes
is
 ItStandard
can be regarded
as the
Tin z
in N small time intervals of length Dt, where
T
N
Dt
n
z (T )  z (0)    i Dt
i 1
9
EXAMPLE12.1(WIENER
PROCESS)
Ex：Initially $25 and time is measured in
years. Mean：25， Standard deviation :1.
At the end of 5 years, what is
mean and Standard deviation?

Our uncertainty about the value of the
variable at a certain time in the future, as
measured by its standard deviation,
increases as the square root of how far
we are looking ahead.
10
GENERALIZED WIENER
PROCESSES(1/3)

A Wiener process, dz, that has been
developed so far has a drift rate (i.e.
average change per unit time) of 0 and
Drift rate →DR , variance rate →VR
a variance
rate of 1

DR=0 means that the expected value of
z at any future time is equal to its
current value.
DR=0 , VR=1

VR=1 means that the variance of the
11
GENERALIZED WIENER
PROCESSES (2/3)

A generalized Wiener process for a
variable x can be defined in terms of dz as
dx = a dt + b dz
DR
VR
12
GENERALIZED WIENER
PROCESSES(3/3)

In a small time interval Δt, the change Δx
in the value of x is given by equations
Dx  aDt  b  Dt
Mean of Δx is
Variance of Δx is
aDt
b 2 Dt
Standard deviation of Δx is
b Dt
13
EXAMPLE 12.2
Follow a generalized Wiener process
1. DR=20 (year) VR=900(year)
2. Initially , the cash position is 50.
3. At the end of 1 year the cash position
will have a normal distribution with a
mean of ★★ and standard deviation
of ●●
ANS:★★=70, ●●=30
15
ITÔ PROCESS

Itô Process is a generalized Wiener
process in which the parameters a
and b are functions of the value of
the underlying variable x and time t.
dx=a(x,t) dt+b(x,t) dz

The discrete time equivalent
is only true in the limit as Dt tends
Dtox zero
a( x, t )Dt  b( x, t )  Dt
16
12.3 THE PROCESS FOR
STOCKS

The assumption of constant
expected drift rate is inappropriate
and needs to be replaced by
assumption that the expected
reture is constant.

This means that in a short interval
of time,Δt, the expected increase in
S is μSΔt.
A stock price does exhibit volatility.

15
AN ITO PROCESS FOR STOCK
PRICES

where m is the expected return and s is the
volatility.
dS  mS dt  sS dz
dS
 mdt  sdz
S

The discrete time equivalent is
DS  mSDt  sS  Dt
16
EXAMPLE
dS 12.3
mS dt  sS dz


Suppose m= 0.15, s= 0.30, then
dS
dS
 mdt  sdz
 0.15dt  0.3dz
S
S
DS
 0.15Dt  0.3  Dt
S
Consider a time interval of 1

week(0.0192)year, so that Dt =0.0192
ΔS=0.00288 S + 0.0416 S
17
MONTE CARLO
SIMULATION


MCSDofS astochastic
mSDt  sS process
 Dt is a procedure
for sampling random outcome for the
process.
DS m=
0.0.14,
0014s=
S 0.2,
0.02
S Dt = 0.01 then
Suppose
and
The first time period(S=20 =0.52 ):
 DS=0.0014*20 +0.02*20*0.52=0.236
The second time period：
 DS'=0.0014*20.236
18
MONTE CARLO
SIMULATION – ONE PATH
Week
Stock Price at
Random
Start of Period Sample for

Change in Stock
Price, DS
0
20.00
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
DS  mSDt  sS  Dt
19
12.4 THE PARAMETERS


We do not have to concern ourselves with the
determinants of μin any detail because the value
of a derivative dependent on a stock is, in
general, independent of μ.
.*.

μ、σ
We will discuss procedures for estimating σ in
Chaper 13
22
12.5 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 )
dx=a(x,t)dt+b(x,t)dz
 Itô's lemma
a functions
G of x
G
Gshows
1  2G that
G
2
dG

(
a


b
)dt  bdz
and t follows the process
2
X
t
2 X
X
21
DERIVATION OF ITÔ'S
LEMMA(1/2)
IfDx is a small change in x and
D G is the resulting small change in
G
dG
DG 
DX

dX
dG
1 d 2G
1 d 3G 3
2
DG 
Dx 
Dx 
Dx  ...
2
3
dX
2 dx
6 dx
G
G
DG 
Dx 
Dy
x
y
Taylor
series
G
G
1  2G
 2G
1  2G
2
2
DG 
Dx 
Dy 
D
x

D
x
D
y

D
y
 ...
2
2
x
y
2 x
xy
2 y
G
G
dG 
dx 
dy
x
y
22
DERIVATION OF ITÔ'S
LEMMA(2/2)

A Taylor's series expansion of G (x, t)
gives
dx  a ( x, t ) dt  b( x, t ) dz
G
G
 2G
2
DG 
Dx 
Dt  ½
D
x
x
t
x 2
 2G
 2G
2

Dx Dt  ½
D
t

2
xt
t
23
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
 G 2
DG 
Dx 
Dt  ½
Dx
2
x
t
x
because Dx has a component which is
2
of order Dt
24
SUBSTITUTING FOR
ΔX
Suppose
dx  a ( x, t )dt  b( x, t )dz
so that
Dx = a Dt + b  Dt
Then ignoring terms of higher order tha n Dt
 G
 G
 2G 2 2
DG 
Dx 
Dt  ½
b  Dt
2
 x
 t
 x
25
2
E ΔT
THE
TERM
Since   f (0,1), E ( )  0
E ( 2 )  [ E ( )]2  1
E ( 2 )  1
It follows that E ( Dt )  Dt
2
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
26
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 and t
G
G
 2G 2 2 
G
dG  
mS 
½
s S dt 
s S dz
2
t
S
S
S

27
APPLICATION TO
FORWARD CONTRACTS
F0  S 0 e rT
F  Se r (T t )
F
 e r (T t )
G
G
 2G 2 2 
G
S

dG  
mS 
½
s
S
dt

s S dz
2


S

t

S

S


2F
0
2
S
F
  rSe r (T t )
t
dF  e r (T t ) mS  rSe r (T t ) dt  e r (T t )sSdz
dF  ( m  r ) Fdt  sFdz


28
THE LOGNORMAL
PROPERTY

We define：
G  ln S
G 1  2G
1 G
 , 2  2,
0
S S S
S t

s2 
dt  s dz
dG   m 
2 

G
G
 2G 2 2 
G


dG  
mS 
½
s S dt 
s S dz
2
t
S
S
S

29
THE LOGNORMAL
PROPERTY

s2 
dt  s dz
dG   m 
2 


s2
2 
ln ST  ln S 0 ~  ( m  )T , s T 
2



s2
2 
ln ST ~  ln S 0  ( m  )T , s T 
2



s T
The standard deviation of the logarithm
of the stock price is
30