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Chapter 20
Brownian Motion and
Itô’s Lemma
1
Introduction
 Stock and other asset prices are commonly assumed to follow a
stochastic process called geometric Brownian motion.
 Given that a stock price follows geometric Brownian motion, we
want to characterize the behavior of a claim that has a payoff
dependent upon the stock price.

We will discuss Ito’s Lemma, which permits us to study the
process followed by a claim that is a function of the stock
price.
2
The Black-Scholes Assumption
about Stock Prices
 The original paper by Black and Scholes begins by assuming that
the price of the underlying asset follows a process like the
following:
where






dS (t )
  dt   dZ (t )
S (t )
(20.1)
S(t) is the stock price,
dS(t) is the instantaneous change in the stock price,
 is the continuously compounded expected return on the
stock,
 is the continuously compounded standard deviation
(volatility),
Z(t) is a normally distributed random variable that follows a
process called Brownian motion.
dZ(t) is the change in Z(t) over a short period of time.
3
The Black-Scholes Assumption
about Stock Prices

A stock obeying equation (20.1) is said to follow a process
called geometric Brownian motion.

Expressions like equation (20.1) are called stochastic
differential equations.

There are 2 important implications of equation (20.1):
1.
Suppose the stock price now is S(0). If the stock price
follows equation (20.1), the distribution of S(T) is
lognormal, i.e.,
ln [S (T)] ~ N (ln [S (0)] + [ – 0.52]T, 2T)
2.
Geometric Brownian motion allows us to describe the
path the stock price takes in getting to a terminal point.
4
A Description of
Stock Price Behavior
 The normal distribution provides an unrealistic description of the
stock price. However, normality can be a plausible description of
continuously compounded returns.

If the continuously compounded return is R(0,T), the price is
S(0) eR(0,T), which is always positive.
 It can be reasonable to assume that over very short periods of
time effective stock returns, S(t + h)/S(t), are normally
distributed.
5
A Description of
Stock Price Behavior
 Suppose that


h, the length of each time period, is small,
the return on the stock, rh, is normally distributed with
mean and variance h and 2h.
 Thus,
S(t + h) = S(t)(1 + rh)
(20.2)
with
rh = ~ N (h, 2h)
6
A Description of
Stock Price Behavior
 The continuously compounded return from 0 to T is
ln[S (T ) / S (0)]   i 1 ln(1  rih )
n
 The logarithm of a normal random variable is not normal.
However, as h  0, the continuously compounded return from 0
to T is normal. Therefore, S(T) tends toward lognormality.
7
Brownian Motion
 Brownian motion is a random walk occurring in continuous time,
with movements that are continuous rather than discrete.

A random walk can be generated by flipping a coin each
period and moving one step, with the direction determined
by whether the coin is heads or tails.

To generate Brownian motion, we would flip the coins
infinitely fast and take infinitesimally small steps at each
point.
8
Brownian Motion
 Let Z(t) represent the value of the random walk—the cumulative
sum of all the moves—after t periods.
 Technically, Brownian motion is a random walk with the following
characteristics:

Z(0) = 0.

Z(t + s) – Z (t) is normally distributed with mean 0 and variance
s.

Z(t + s1) – Z(t) is independent of Z(t) – Z(t – s2), where s1, s2 >0.
That is, nonoverlapping increments are independently
distributed.

Z(t) is continuous.
9
Brownian Motion
 We can think of Brownian motion being approximately
generated from the sum of independent binomial draws with
mean 0 and variance h.
 Let h get arbitrarily small, and rename h as dt.
Denote the change in Z as dZ(t).
10
Brownian Motion
 Then,
dZ (t )  Y (t ) dt
(20.5)
where Y(t) is a random draw from a binomial distribution, such that Y(t)
is 1 with probability 50%.
 Equation (20.5) says “Over small periods of time, changes in the
value of the process are normally distributed with a variance that
is proportional to the length of the time period.”
11
Brownian Motion
 Since Z(T) is the sum of individual dZ(t)’s, we can write
T
Z (T )  Z (0)   dZ (t )
(20.6)
0
The integral here is called a stochastic integral.
The process Z(t) is also called a diffusion process.
 Among properties of Brownian motion are:


“infinite-crossing property,” and
“infinite variation.”
12
Arithmetic Brownian Motion
 With pure Brownian motion, the expected change in Z is 0, and
the variance per unit time is 1. We can generalize this to allow
an arbitrary variance and a nonzero mean.
dX (t )   dt   dZ (t )
(20.8)
This process is called arithmetic Brownian motion.
  is the instantaneous mean per unit time,

2 is the instantaneous variance per unit time,

the variable X(t) is the sum of the individual changes dX.
X(t) is normally distributed, i.e., X(T) – X(0) ~ N (T, 2T).
13
Arithmetic Brownian Motion
 An integral representation of equation (20.8) is
T
T
0
0
X (T )  X (0)    dt    dZ (t )
 Here are some properties of the process in equation (20.8):

X(t) is normally distributed because it is created by
adding together many normally distributed dX’s.

The random term is multiplied by a scale factor that
enables us to change variance.

The dt term introduces a nonrandom drift into the
process.
14
Arithmetic Brownian Motion
 Arithmetic Brownian motion has several drawbacks:

There is nothing to prevent X from becoming negative, so it
is a poor model for stock prices.

The mean and variance of changes in dollar terms are
independent of the level of the stock price.
 Both of these criticisms will be eliminated with geometric
Brownian motion.
15
The Ornstein-Uhlenbeck Process
 We can incorporate mean reversion by modifying the drift
term:
dX (t )  [  X (t )] dt   dZ (t )
(20.9)
When  = 0, this equation is called an Ornstein-Uhlenbeck
process.

The parameter  measures the speed of the reversion: If 
is large, reversion happens more quickly.

In the long run, we expect X to revert toward .

As with arithmetic Brownian motion, X can still become
negative.
16
Example of Mean-Reverting Process
1
0.8
0.6
0.4
0.2
0
0
1000
2000
3000
4000
5000
6000
-0.2
17
Geometric Brownian Motion
 An equation, in which the drift and volatility depend on the
stock price, is called an Itô process.
Suppose we modify arithmetic Brownian motion to make
the instantaneous mean and standard deviation
proportional to X(t):
dX (t )  a X (t ) dt   X (t )dZ (t )
This is an Itô process that can also be written
dX (t )
 a dt   dZ (t )
X (t )
(20.11)
This process is known as geometric Brownian motion.
18
Geometric Brownian Motion
 The percentage change in the asset value is normally
distributed with instantaneous mean  and instantaneous
variance 2.
 The integral representation for equation (20.11) is
T
T
0
0
X (T )  X (0)    X (t )dt    X (t )dZ (t )
19
Example of Geometric Brownian
Motion
50
45
40
35
30
25
20
15
10
5
0
0
1000
2000
3000
4000
5000
6000
20
Lognormality
 If a variable is distributed in such a way that instantaneous
percentage changes follow geometric Brownian motion, then
over discrete periods of time, the variable is lognormally
distributed.
21
Relative importance of the drift and
noise terms
 Over short periods of time, the character of the Brownian
process is determined almost entirely by the random
component.

Consider the ratio of the per-period standard deviation to the
per-period drift:
X (t ) h


X (t )h  h
The ratio becomes infinite as h approaches dt.
 As the time interval becomes longer, the mean becomes more
important than the standard deviation.
22
Relative importance of the drift and
noise terms
The results in the table hold for both geometric Brownian motion
and arithmetic Brownian motion.
23
Correlated Itô processes
 Let W1(t) and W2(t) be independent Brownian motions.
Then,
Z (t )  W1 (t )
Z ' (t )  W1 (t )  1   2 W2 (t )
(20.16)
This is the Cholesky decomposition.
 The correlation between Z(t) and Z'(t) is
E[ Z (t ) Z ' (t )]  E[W1 (t ) 2 ]  1   2 E[W1 (t )W2 (t )]   t  0
The second term on the right-hand side is 0 because W1(t) and
W2(t) are independent.
24
Multiplication rules
 We can simplify complex terms containing dt and dZ by
using the following “multiplication rules”:
dt  dZ = 0
(dt)2 = 0
(20.17a)
(20.17b)
(dZ)2 = dt
(20.17c)
dZ  dZ' = dt
(20.17d)
The reason behind these multiplication rules is that the
multiplications resulting in powers of dt greater than 1 vanish.
25
The Sharpe Ratio
 If asset i has expected return i, the risk premium is defined as
Risk premiumi = i – r
where r is the risk-free rate.
 The Sharpe ratio for asset i is the risk premium, i – r, per unit
of volatility, i:
Sharpe ratio i 
αi  r
i
(20.18)
26
The Sharpe Ratio
 We can use the Sharpe ratio to compare two perfectly
correlated claims, such as a derivative and its underlying
asset.
 Two assets that are perfectly correlated must have the same
Sharpe ratio, or else there will be an arbitrage opportunity.

Consider the processes for two non-dividend paying
stocks:
dS1 = 1S1dt + 1S1dZ
dS2 = 2S2dt + 2S2dZ
(20.19)
(20.20)
Because the two stock prices are driven by the same dZ,
it must be the case that (1 – r)/ 1 = (2 – r)/ 2.
27
The risk-neutral process
 Suppose the true price process is
dS (t )
 (   )dt  dZ (t )
S (t )
where  is the dividend yield on the stock.
 We can write a risk-neutral version of this process by
subtracting the risk premium,  – r, from the drift,  – :
dS (t )
(20.25)
 (r   )dt   dZ * (t )
S (t )


The probability distribution (dZ*(t)) associated with
the risk-neutral process is said to be the riskneutral measure.
When we switch to the risk-neutral process, the
volatility remains the same.
28
Itô’s Lemma
 Suppose a stock with an expected instantaneous return of ,
dividend yield of , and instantaneous volatility  follows
geometric Brownian motion:
dS(t) = ( – )S(t)dt + S(t)dZ(t)
(20.26)
 C[S(t), t] is the value of a derivative claim that is a function of the
stock price.
How can we describe the behavior of this claim in terms of the
behavior of S?
29
Itô’s Lemma
 Itô’s Lemma (Proposition 20.1)
If C[S(t), t] is a
twice-differentiable function of S(t), then the change
in C is
1
dC ( S , t )  CS dS  CSS (dS ) 2  Ct dt
2
(20.27)
where we recognize the delta-gamma approximation.

30
Itô’s Lemma
 In the case where S follows a Geometric
Brownian motion, we have:
dS  (   ) Sdt  σSdZ
and so
1 2 2


dC  (   ) SCS   S CSS  Ct  dt  σSCS dZ
2


where CS = C/S, CSS = 2C/S2, and Ct = C/t.
 The terms in square brackets are the expected
change in the option price.
31
Itô’s Lemma Application
 Let C(S) = ln(S) so that the value of C is
the logarithm of S.
 Let dS = Sdt + SdZ
 Compute the path for the changes in C,
i.e. compute the expression for dC.
32
Itô’s Lemma Application: solution
 CS = C/S = (ln(S))/S = 1/S.
 CSS = 2C/S2 = (1/S)/S = -1/S2.
 Ct = C/t = 0 because C = ln(S) and is
not a function of t.
 Therefore dC = [ – (1/2)2]dt + dZ
33
Multivariate Itô’s Lemma
 A derivative may have a value depending on more than one
price, in which case we can use a multivariate generalization of
Itô’s Lemma.
 Multivariate Itô’s Lemma (Proposition 20.2)
Suppose we
have n correlated Itô processes:
dSi (t )
  i dt   i dzi ,
Si (t )
i  1, . . ., n
Denote the pairwise correlations as E(dzi  dzj) = i,jdt.
34
Multivariate Itô’s Lemma
 If C(S1, . . ., Sn, t) is a twice-differentiable function of the Si’s, we
have
dC ( S1 , . . ., S n , t )  i 1 CSi dSi 
n
1 n
n
dSi dS j CSi S j  Ct dt

i 1  j 1
2
 The expected change in C per unit time is
1
1 n
n
n
E[dC ( S1 , . . ., S n , t )]  i 1 i Si CSi  i 1  j 1 i j ij Si S j CSi S j  Ct
dt
2
35
Valuing a Claim on Sa
 Suppose a stock with an expected instantaneous return of ,
dividend yield of , and instantaneous volatility  follows
geometric Brownian motion:
dS(t) = ( – )S(t)dt + S(t)dZ(t)
 Now suppose that we also have a claim with a payoff depending
on S raised to some power.
 For example, we may have a claim that pays S(T)2 at time T.
36
Valuing a Claim on Sa
 Proposition 20.3
The value at time 0 of a claim paying
S(T)a—the prepaid forward price—is
1
[ a ( r  )  a ( a 1) 2 ]T
a
2
F0P,T [ S (T ) a ]  e  rT S (0) e
(20.29)
The forward price for S(T)a is
1
[ a ( r  )  a ( a 1) 2 ]T
a
2
F0,T [ S (T ) a ]  S (0) e
(20.30)
37
The process followed by Sa
 Consider a claim maturing at time T that pays C[S(T),
T] = S(T)a.
 We can use Itô’s Lemma to determine the process
followed by Sa.
 Give it a try, by using:
1
dC ( S , t )  CS dS  CSS (dS ) 2  Ct dt
2
38
The process followed by Sa
 We get:
dS a 
1
2

a
(



)

a
(
a

1
)

dt  adZ
a


S
2


(20.31)

Thus, Sa follows geometric Brownian motion with drift
a ( – ) + 1/2a (a – 1)2 and risk adZ.

If  is the expected return for S, the expected return of a
claim with price Sa will be
a ( – r) + r
(20.32)
and the risk premium will be a ( – r). This is necessary
to have the same Sharpe ratios.
39
Examples
 Claims on S:
Suppose a = 1. Equation (20.29) then gives us
the prepaid forward price on a stock:
V (0)  S (0)e T
 Claims on S0:
If a = 0, the claim does not depend on the
stock price. Since S0 = 1, it is a bond. Setting a = 0 gives us
the price of a T-period pure discount bond:
V (0)  e  rT
40
Examples
 Claims on S2:
When a = 2, the claim pays S(T)2. From
equation (20.30), the forward price is
F0,T ( S )  e S (0) e
2
rT
2
 S (0) e
2
[  r  2  2 ]T
2 ( r  )T  2T
e
(20.35)
2  2T
 [ F0,T ( S )] e
Thus, the forward price on the squared stock price is the squared
forward price times a variance term.
41
Examples
If a = –1, the claim pays 1/S. Using
equation (20.30), we get
 Claims on 1/S:
F0,T (1/ S )  [1/ S (0)]e
(  r )T  2T
e
As with the squared security, the forward price is increasing in
volatility.

The payoffs for both the S2 and 1/S securities are
convex. Therefore, according to Jensen’s inequality, the
price is higher when the asset price is risky than when it
is certain.
42
Valuing a claim on SaQb
 Suppose a claim pays S(T)aQ(T)b, where S follows
dS
 ( S   S )dt   S dZ S
S
(20.36)
and Q follows
dQ
 ( Q   Q )dt   Q dZ Q
Q
(20.37)
where
dZ S dZ Q   dt
43
Valuing a claim on SaQb
 Proposition 20.4
Suppose that the forward prices for Sa
and Qb are given by Proposition 20.3. Then the forward
price for SaQb is
Ft ,T (S aQb )  Ft ,T (S a ) Ft ,T (Qb )e
ab S  Q (T t )
The forward price for SaQb is the product of those two forward
prices times a factor that accounts for the covariance between the
two assets.
 The variance of SaQb is given by
a 2 S2  b2 Q2  2ab S Q
44
Valuing a claim on SaQb
 The squared security, S2, is a special case of Proposition 20.4.
When S = Q, a = b = 1, and  = 1, the covariance term becomes
ab S Q   S2
This gives us the same result as equation (20.35) for the forward price
for a squared stock.
45
Valuing a claim on
a
b
SQ
 Proposition 20.4 can be generalized.
Suppose there are n stocks, each of which follows the
process
dSi
 ( i   i )dt   i dzi
Si
(20.40)
where dzidzj = ijdt. Let
V (t )  i 1 Siai
n
(20.41)
The forward price for V is then
F0,T (V )  i 1[ F0,T ( Si )] e
n
ai
i1  ji1 ij i j ai a j
n1
n
(20.42)
46
Jumps in the Stock Price
 One objection to the Brownian process as a model of the stock
price is that Brownian paths are continuous—there are no
discrete jumps in the stock price.
In practice, asset prices occasionally seem to jump (e.g., on
October 19, 1987).
One way to model such jumps is by using the Poisson
distribution mixed with a standard Brownian process.
47
Jumps in the Stock Price
 Let q(t) represent the cumulative jump and dq the change in the
cumulative jump.

When there is no jump, dq = 0.

When there is a jump, we let the random variable Y denote
the magnitude of the jump, and k = E(Y) – 1 is then the
expected percentage change in the stock price.
 If  is the expected number of jumps per unit time over an
interval dt, then
Prob(jump) = dt
Prob(no jump) = 1 – dt
48
Jumps in the Stock Price
 We can write the stock price process as
dS(t)/S(t) = ( – k)dt + dZ + dq
(20.43)
where
0
if there is no jump
Y-1
if there is a jump
dq =
and E(dq) = kdt.
 When no jump is occurring, the stock price S evolves as geometric
Brownian motion. When the jump occurs, the new stock price is
YS.
49
Example of Jumps in the Process
80
70
60
50
40
30
20
10
0
0
1000
2000
3000
4000
5000
6000
50
Jumps in the Stock Price
 Proposition 20.5
Suppose an asset follows equation (20.43).
If C(S, t) is a twice continuously differentiable function of the
stock price, the process followed by C is
1
dC ( S , t )  CS dS  CSS 2 S 2 dt  Ct dt  EY [C ( SY , t )  C ( S , t )]
2
(20.44)
 The last term in the equation is the expected change in the
option price conditional on the jump times the probability of the
jump.
51