Outline
The Markov property
Wiener Process
Stock price process
Correlated processes
Itô’s lemma
Wiener Processes and Itô’s Lemma
Haipeng Xing
Department of Applied Mathematics and Statistics
Haipeng Xing, AMS320, Stony Brook University
Wiener Processes and Itô’s Lemma
Outline
The Markov property
Wiener Process
Stock price process
Correlated processes
Itô’s lemma
Outline
1
The Markov property
2
Wiener Process
3
The process for a stock price
4
Correlated processes
5
Itô’s lemma
Haipeng Xing, AMS320, Stony Brook University
Wiener Processes and Itô’s Lemma
Outline
The Markov property
Wiener Process
Stock price process
Correlated processes
Itô’s lemma
The Markov property
A Markov process is a particular type of stochastic process where
only the current 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.
Stock prices are usually assumed to follow a Markov process. The
Markov property implies that the probability distribution of the price
at any particular future time does not depend on the particular path
followed by the price in the past.
The Markov property of stock prices is consistent with the weak
form of market efficiency, which states that the present price of a
stock impounds all the information contained in a record of past
prices. — If the weak form of market efficiency were not true,
technical analysts could make above-average returns by studying the
history of stock prices.
The competition in the market tends to ensure that weak-form of
market efficiency and the Markov property hold.
Haipeng Xing, AMS320, Stony Brook University
Wiener Processes and Itô’s Lemma
Outline
The Markov property
Wiener Process
Stock price process
Correlated processes
Itô’s lemma
Wiener Process
Let N (µ, ⌫) be a normal distribution with mean µ and variance ⌫.
A variable z follows a Wiener process if it satisfies the following two
properties:
The change
z during a small period of time t is
p
z=✏
t,
where ✏ ⇠ N (0, 1).
(1)
p
The values of zi = ✏i
ti (i = 1, 2) for any two di↵erent short
intervals of time t1 and t2 are independent.
It follows from the first property that z ⇠ N (0, t). The second
property implies that z follows a Markov process.
Haipeng Xing, AMS320, Stony Brook University
Wiener Processes and Itô’s Lemma
Outline
The Markov property
Wiener Process
Stock price process
Correlated processes
Itô’s lemma
Wiener process
In ordinary calculus, it is usual to proceed from small changes to the
limit as the small changes become closer to zero. We use similar
notational conventions in stochastic calculus. So when we refer to
dz as a Wiener process, we mean that it has the properties for z
given above in the limit as t ! 0.
Figure 1: A Wiener process is obtained when
2014; Figure 14.1).
t ! 0 in equation (1) (Hull,
Haipeng Xing, AMS320, Stony Brook University
Wiener Processes and Itô’s Lemma
Outline
The Markov property
Wiener Process
Stock price process
Correlated processes
Itô’s lemma
Generalized Wiener process
The mean change per unit time for a stochastic process is known as
the drift rate and the variance per unit time is known as the variance
rate.
The basic Wiener process, dz, has a drift rate of 0 and a variance
rate of 1. —— The drift rate of 0 means that the expected value of
z at any future time is equal to its current value. The variance rate
of 1 means that the variance of the change in z in a time interval of
length T equals T .
A generalized Wiener process for a variable x can be defined in
terms of dz as
dx = adt + bdz,
(2)
where a and b are constant.
Haipeng Xing, AMS320, Stony Brook University
Wiener Processes and Itô’s Lemma
Outline
The Markov property
Wiener Process
Stock price process
Correlated processes
Itô’s lemma
Generalized Wiener process
In a small time interval
by equation (2) as
t, the change
p
x = a t + b✏
t,
x in the value of x is given
where ✏ ⇠ N (0, 1).
Figure 2: Generalized Wiener process (2) with a = 0.3 and b = 1.5 (Hull,
2014; Figure 14.2).
Haipeng Xing, AMS320, Stony Brook University
Wiener Processes and Itô’s Lemma
Outline
The Markov property
Wiener Process
Stock price process
Correlated processes
Itô’s lemma
Itô process
In an Itô process, the drift rate and the variance rate are functions of
time
dx = a(x, t)dt + b(x, t)dz.
(3)
Both the expected drift rate and variance rate of an Itô process are
liable to change over time. In a small time interval between t and
t + t, the variable changes from x to x + x, where
p
x ⇡ a(x, t) t + b(x, t)✏
t.
Note that the process in equation (3) is Markov because the change
in x at time t depends only on the value of x at time t, not on its
history. A non-Markov process could be defined by letting a and b in
equation (3) depend on values of x prior to time t.
Haipeng Xing, AMS320, Stony Brook University
Wiener Processes and Itô’s Lemma
Outline
The Markov property
Wiener Process
Stock price process
Correlated processes
Itô’s lemma
The process for a stock price
For a stock price we can assume that its expected return or
percentage change in a short period of time is constant (not its
expected actual change).
We can also assume that the standard deviation of the change in a
short period of time t is proportional to the level of the stock price.
The above assumptions lead to a model of stock price behavior
dS
= µdt + dz,
S
(4)
where µ is the stock’s expected rate of return and is the volatility
of the stock price. Model (4) is known as geometric Brownian
motion.
The model in (4) represents the stock price process in the real world.
In a risk-neutral world, µ equals the risk-free rate r.
Haipeng Xing, AMS320, Stony Brook University
Wiener Processes and Itô’s Lemma
Outline
The Markov property
Wiener Process
Stock price process
Correlated processes
Itô’s lemma
The process for a stock price
The discrete-time version of model (4) is
p
S = µS t + S✏
t,
(5)
where S is the change in the stock price S in a small time interval
t, and ✏ ⇠ N (0, 1).
Equation (5) suggests that
S
⇠ N (µ t,
S
2
t).
One can simulate a path of stock price S based on equaton (5).
Haipeng Xing, AMS320, Stony Brook University
Wiener Processes and Itô’s Lemma
Outline
The Markov property
Wiener Process
Stock price process
Correlated processes
Itô’s lemma
Correlated processes
Suppose dz1 and dz2 are Wiener processes with correlation ⇢. Then
p
p
z1 = ✏1
t,
z2 = ✏2
t,
where ✏1 and ✏2 are random samples from a bivariate standard
normal distribution where corrleaton is ⇢.
Standard normal variables ✏1 and ✏2 with correlation ⇢ can be
sampled by setting
p
✏1 = u and ✏2 = ⇢u + 1 ⇢2 v,
where u ⇠ N (0, 1), v ⇠ N (0, 1), and u and v are independent.
Haipeng Xing, AMS320, Stony Brook University
Wiener Processes and Itô’s Lemma
Outline
The Markov property
Wiener Process
Stock price process
Correlated processes
Itô’s lemma
Itô’s lemma
In stochastic calculus, if we know the stochastic process followed by
x, Itô’s lemma can tell us the stochastic process followed by some
function G(x, t).
Since the price of a derivative is a function of the stochastic
variables underlying the derivative and time, Itô’s lemma plays an
important role in the analysis of derivatives.
Consider a stochastic process that is represented by
dx = a(x, t)dt + b(x, t)dz.
Then the change of x in a small time interval t can be
approximated by
p
x = a(x, t) t + b(x, t)✏
t.
(6)
(7)
Haipeng Xing, AMS320, Stony Brook University
Wiener Processes and Itô’s Lemma
Outline
The Markov property
Wiener Process
Stock price process
Correlated processes
Itô’s lemma
Derivation of Itô’s lemma
Recall that a Taylor’s series expansion of G(x, t) gives
G
=
@G
@G
1 @2G
x+
t+
( x)2
2
@x
@t
2 @x
@2G
1 @2G
+
( x)( t) +
( t)2 + . . .
2
@x@t
2 @t
In ordinary calculus, we ignore terms of higher order than
obtain that
@G
@G
G=
x+
t.
@x
@t
In stochastic calculus, this becomes
because
(8)
t and
@G
@G
1 @2G
G=
x+
t+
( x)2 ,
2
@x
@t
2 @x
p
x has a component which is of order
t.
(9)
Haipeng Xing, AMS320, Stony Brook University
Wiener Processes and Itô’s Lemma
Outline
The Markov property
Wiener Process
Stock price process
Correlated processes
Itô’s lemma
Derivation of Itô’s lemma
Substituting (7) into (9) yields that
⌘
@G
@G
1 @2G ⇣ 2
2
3/2
2 2
G=
x+
t+
a ( t) + 2ab✏( t) + b ✏ t
@x
@t
2 @x2
Note that ✏ ⇠ N (0, 1), hence E(✏) = 0 and E(✏2 ) = 1. Ignoring the
term of higher order than t yields that
@G
@G
1 @2G 2
x+
t+
b t.
@x
@t
2 @x2
t ! 0, we obtain
G=
By taking limit
@G
@G
1 @2G 2
dx +
dt +
b dt.
(10)
@x
@t
2 @x2
Substituting dx = a(x, t)dt + b(x, t)dz into (10) yields the Itô’s
lemma
⇣ @G
@G 1 2 @ 2 G ⌘
@G
dG =
+a
+ b
dt
+
b
dz.
(11)
@t
@x
2 @x2
@x
dG =
Haipeng Xing, AMS320, Stony Brook University
Wiener Processes and Itô’s Lemma
Outline
The Markov property
Wiener Process
Stock price process
Correlated processes
Itô’s lemma
Itô’s lemma: Examples
Consider the geometric Brownian motion for stock price movements,
dS = µSdt + Sdz.
Let G(S, t) be a function of S and t, it follows from Itô’s lemma that
dG(S, t) =
⇣ @G
@G 1
+ µS
+
@t
@S
2
2
G⌘
@G
S
dt + S
dz.
2
@S
@S
2@
2
Consider the log of a stock price, G(S, t) = log S, we have
⇣
1 2⌘
dG = µ
dt + dz.
2
Consider the forward price of a stock for a contract maturing at time
T , G = Ser(T t) , we have
dG = (µ
r)Gdt + Gdz.
Haipeng Xing, AMS320, Stony Brook University
Wiener Processes and Itô’s Lemma