11.1
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Model of the
Behavior
of Stock Prices
Chapter 11
11.2
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Categorization of Stochastic
Processes
? Discrete time; discrete variable?
Discrete time; continuous variable?
Continuous time; discrete variable?
Continuous time; continuous variable
11.3
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
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
derivatives
11.4
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Markov Processes (See pages 216-7)
? 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 assume that stock prices follow
Markov processes
11.5
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
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
clearly consistent with weak-form market
efficiency
11.6
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
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.
11.7
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Questions
? What is the probability distribution of the
stock price at the end of 2 years??
? years??
? years??
dt years?
Taking limits we have defined a
continuous variable,continuous time
process
11.8
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
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
11.9
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
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.
11.10
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
A Wiener Process (See pages 218)
? 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,
2,The values of dz for any 2 different (non-
overlapping) periods of time are independent
( 0,1 ) f r o m d r a w i n g r a n d o m a is w h e r e f?d??d tz
11.11
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
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
11.12
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
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
11.13
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Generalized Wiener Processes
(See page 220-2)
? 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
11.14
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Generalized Wiener Processes
(continued)
The variable x follows a generalized
Wiener process with a drift rate of a
and a variance rate of b2 if
dx=adt+bdz
11.15
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
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
tbtax d??d?d
b T
11.16
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
The Example Revisited
? A stock price starts at 40 and 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
11.17
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Ito Process (See pages 222)
? 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
ttxbttxax d??d?d ),(),(
11.18
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
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
absolute change in a short period of time
? 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
11.19
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
An Ito Process for Stock Prices
(See pages 222-3)
where m is the expected return s
is the volatility.
The discrete time equivalent is
dS Sdt Sdz? ?m s
tStSS d?s?dm?d
11.20
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
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
???d SSS 02.00 0 1 4.0
11.21
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Monte Carlo Simulation – One Path
(See Table 11.1)
P e ri o d
S to ck Pri ce a t
S ta rt o f Pe ri o d
R a n d o m
S a m p le fo r ?
C h a n g e in Sto ck
P ri ce,? 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
11.22
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Ito’s Lemma (See pages 226-227)
? 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 and time,Ito’s
lemma plays an important part in the
analysis of derivative securities
11.23
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Taylor Series Expansion
? A Taylor’s series expansion of G(x,t)
gives
??d
?
?
?dd
??
?
?
d
?
?
?d
?
?
?d
?
?
?d
2
2
22
2
2
2
t
t
G
tx
tx
G
x
x
G
t
t
G
x
x
G
G
?
?
11.24
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Ignoring Terms of Higher Order
Than dt
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
o r d e r of
is w h i c hc o m p o n e n t a has b e c a u s e
?
b e c o m e s t h i s c a l c u l u s s t o c h a s t i c In
h a v e w ec a l c u l u so r d i n a r y In
2
2
2
11.25
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Substituting for dx
tb
x
G
t
t
G
x
x
G
G
t
tbtax
dztxbdttxadx
d?
?
?
?d
?
?
?d
?
?
?d
d
d?dd
??
22
2
2
?
t h a n o r d e r h i g h e r of t e r m s i g n o r i n g T h e n
+ =
t h a t so
),(),(
S u p p o s e
11.26
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
The ?2?t Term
tb
x
G
t
t
G
x
x
G
G
tt
ttE
E
EE
E
d
?
?
?d
?
?
?d
?
?
?d
dd
d?d?
??
????
??f??
2
2
2
2
2
22
2
1
)(
1)(
1)]([)(
0)()1,0(
H e n c e i g n o r e d, be
c a n a n d to alp r o p o r t i o n is of v a r i a n c e T h e
t h a t f o l l o w s It
S i n c e
2
11.27
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Taking LimitsT a king l i m i ts ?
Su b stituti n g
W e o b ta i n ?
T h i s i s Ito ' s L e m m a
dG
G
x
dx
G
t
dt
G
x
b dt
dx a dt b dz
dG
G
x
a
G
t
G
x
b dt
G
x
b dz
? ? ?
? ?
? ? ?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
2
2
2
2
2
2
11.28
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Application of Ito’s Lemma
to a Stock Price Process
dzS
S
G
dtS
S
G
t
G
S
S
G
dG
tSG
zdSdtSSd
?
a n d of f u n c t i o n a F o r
is p r o c e s s p r i c e s t o c k T h e
s
?
?
?
?
?
?
?
?
?
?
?
s
?
?
?
?
?
?m
?
?
?
s?m?
22
2
2
11.29
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Examples
dzdtdG
SG
dzGdtGrdG
eSG
T
tTr
2.
t i m e at m a t u r i n g
c o n t r a c t a f o r s t o c k a of p r i c e f o r w a r d T h e 1.
s?
?
?
?
?
?
?
?
? s
?m?
?
s??m?
?
?
2
ln
)(
2
)(