Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.1
Model of the
Behavior
of Stock Prices
Chapter 10
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.2
Categorization of Stochastic
Processes
Discrete time; discrete variable
Discrete time; continuous variable
Continuous time; discrete variable
Continuous time; continuous variable
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.3
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.4
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.5
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.6
Example of a Discrete Time
Continuous Variable Model
A stock price 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,
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.7
Questions
What is the probability distribution of the
change in stock price over/during
2 years?
years?
years?
Dt years?
Taking limits we have defined a continuous
variable,continuous time process
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.8
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.9
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,(You can say that the STD
is 10 per square root of years)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.10
A Wiener Process (See pages 220-1)
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.,where? is a random drawing
from f(0,1),
2,The values of Dz for any 2 different (non-
overlapping) periods of time are independent
tz D?D?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.11
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 T
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.12
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.13
Generalized Wiener Processes
(See page 221-4)
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.14
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 = a dt + b dz
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.15
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
D D Dx a t b t
b T
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.16
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.17
Ito Process (See pages 224-5)
In an Ito 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 only true in the limit as Dt tends to
zero
D D Dx a x t t b x t t(,) (,)?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.18
Why a Generalized Wiener Process
is not Appropriate for Stocks
For a stock price we can conjecture that
its expected proportional change in a
short period of time remains constant
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.19
An Ito Process for Stock Prices
(See pages 225-6)
where m is the expected return,s is the
volatility.
The discrete time equivalent is
dS Sdt Sdzm s
D D DS S t S tm s?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.20
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
SSS 02.00 0 1 4.0D
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.21
Monte Carlo Simulation – One Path
(continued,See Table 10.1)
Pe ri o d
Sto c k Pri c e a t
Sta rt o f Pe ri o d
Ran d o m
Sa m p l e f o r?
Cha n g e i n St o c k
Pri c e,D S
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.22
Ito’s Lemma (See pages 229-231)
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.23
Taylor Series Expansion
A Taylor’s series expansion of
G (x,t ) gives
D D D D
D D D
G
G
x
x
G
t
t
G
x
x
G
x t
x t
G
t
t
2
2
2
2 2
2
2?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.24
Ignoring Terms of Higher Order
Than Dt
In or dinar y cal cul us w e g et
sti c c alc u l us w e g et
be cause has a co mp on en t w h i ch i s of or de r
In st ocha
D D D
D D D D
D D
G
G
x
x
G
t
t
G
G
x
x
G
t
t
G
x
x
x t
2
2
2
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.25
Substituting for Dx
Suppose
(,) (,)
so that
= +
T hen ig nor i ng ter ms of hi g her or der than
dx a x t dt b x t dz
x a t b t
t
G
G
x
x
G
t
t
G
x
b t
D D D
D
D D D D
2
2
2 2
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.26The?2Dt Term
tb
x
G
t
t
G
x
x
G
G
tt
ttE
E
EE
E
D?D?D?D
DD
D?D
2
2
2
22
2
2
22
2
1
H e n c e i g n or e d,be
can a n d t oalpr op or t i on is of v a r i a n c eT h e
)( t h a t f ol l ow sI t
1)(
1)]([)(
0)(,)1,0( S i n c e
f?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.27
Taking Limits
dzb
x
G
dtb
x
G
t
G
a
x
G
dG
dzbdtadx
dtb
x
G
dt
t
G
dx
x
G
dG
o b t a i n We
ngSu b s t i t u t i
l i m i t s T a k i n g
2
2
2
2
2
2
This is Ito’s Lemma.
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.28
Application of Ito’s Lemma
to a Stock Price Process
T he s toc k pr i c e pr oc es s i s
F or a f unc ti on o f &
d S S dt S d z
G S t
dG
G
S
S
G
t
G
S
S dt
G
S
S dz
m s
m
s
s
2
2
2 2
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.29
Examples
1,T he f or w ar d p r i ce of a sto ck f or a co n tr act
m atu r i ng at ti m e
e
2,
T
G S
dG r G dt G dz
G S
dG dt dz
r T t
( )
( )
ln
m s
m
s
s
2
2
Tang Yincai,? 2003,Shanghai Normal University
10.1
Model of the
Behavior
of Stock Prices
Chapter 10
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.2
Categorization of Stochastic
Processes
Discrete time; discrete variable
Discrete time; continuous variable
Continuous time; discrete variable
Continuous time; continuous variable
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.3
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.4
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.5
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.6
Example of a Discrete Time
Continuous Variable Model
A stock price 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,
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.7
Questions
What is the probability distribution of the
change in stock price over/during
2 years?
years?
years?
Dt years?
Taking limits we have defined a continuous
variable,continuous time process
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.8
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.9
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,(You can say that the STD
is 10 per square root of years)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.10
A Wiener Process (See pages 220-1)
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.,where? is a random drawing
from f(0,1),
2,The values of Dz for any 2 different (non-
overlapping) periods of time are independent
tz D?D?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.11
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 T
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.12
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.13
Generalized Wiener Processes
(See page 221-4)
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.14
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 = a dt + b dz
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.15
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
D D Dx a t b t
b T
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.16
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.17
Ito Process (See pages 224-5)
In an Ito 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 only true in the limit as Dt tends to
zero
D D Dx a x t t b x t t(,) (,)?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.18
Why a Generalized Wiener Process
is not Appropriate for Stocks
For a stock price we can conjecture that
its expected proportional change in a
short period of time remains constant
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.19
An Ito Process for Stock Prices
(See pages 225-6)
where m is the expected return,s is the
volatility.
The discrete time equivalent is
dS Sdt Sdzm s
D D DS S t S tm s?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.20
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
SSS 02.00 0 1 4.0D
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.21
Monte Carlo Simulation – One Path
(continued,See Table 10.1)
Pe ri o d
Sto c k Pri c e a t
Sta rt o f Pe ri o d
Ran d o m
Sa m p l e f o r?
Cha n g e i n St o c k
Pri c e,D S
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.22
Ito’s Lemma (See pages 229-231)
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.23
Taylor Series Expansion
A Taylor’s series expansion of
G (x,t ) gives
D D D D
D D D
G
G
x
x
G
t
t
G
x
x
G
x t
x t
G
t
t
2
2
2
2 2
2
2?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.24
Ignoring Terms of Higher Order
Than Dt
In or dinar y cal cul us w e g et
sti c c alc u l us w e g et
be cause has a co mp on en t w h i ch i s of or de r
In st ocha
D D D
D D D D
D D
G
G
x
x
G
t
t
G
G
x
x
G
t
t
G
x
x
x t
2
2
2
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.25
Substituting for Dx
Suppose
(,) (,)
so that
= +
T hen ig nor i ng ter ms of hi g her or der than
dx a x t dt b x t dz
x a t b t
t
G
G
x
x
G
t
t
G
x
b t
D D D
D
D D D D
2
2
2 2
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.26The?2Dt Term
tb
x
G
t
t
G
x
x
G
G
tt
ttE
E
EE
E
D?D?D?D
DD
D?D
2
2
2
22
2
2
22
2
1
H e n c e i g n or e d,be
can a n d t oalpr op or t i on is of v a r i a n c eT h e
)( t h a t f ol l ow sI t
1)(
1)]([)(
0)(,)1,0( S i n c e
f?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.27
Taking Limits
dzb
x
G
dtb
x
G
t
G
a
x
G
dG
dzbdtadx
dtb
x
G
dt
t
G
dx
x
G
dG
o b t a i n We
ngSu b s t i t u t i
l i m i t s T a k i n g
2
2
2
2
2
2
This is Ito’s Lemma.
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.28
Application of Ito’s Lemma
to a Stock Price Process
T he s toc k pr i c e pr oc es s i s
F or a f unc ti on o f &
d S S dt S d z
G S t
dG
G
S
S
G
t
G
S
S dt
G
S
S dz
m s
m
s
s
2
2
2 2
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
10.29
Examples
1,T he f or w ar d p r i ce of a sto ck f or a co n tr act
m atu r i ng at ti m e
e
2,
T
G S
dG r G dt G dz
G S
dG dt dz
r T t
( )
( )
ln
m s
m
s
s
2
2
