Chapter 20 Brownian Motion and Itô’s Lemma. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 20-2 Introduction Stock and other asset prices

Chapter 20

Brownian Motion and Itô’s Lemma

Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 20-2

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 Itô’s Lemma, which permits us to

study the process followed by a claim that is a function of the stock price


)( )(

)(tdZdt

tS

tdS σα +=

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

(20.1)

where

S(t) is the stock price dS(t) is the instantaneous change in the stock price α is the continuously compounded expected return on the stock σ 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


The Black-Scholes Assumption About Stock Prices (cont’d)

• 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) 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.

Geometric Brownian motion allows us to describe the path the stock price takes in getting to a terminal point

In[S(T )] ~ N(In[S(O)]+[α −0.5σ 2 ]T,σ 2T )


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


A Description of Stock Price Behavior (cont’d)

• Suppose that h, the length of each time period, is small,

the return on the stock, rh, is normally distributed with mean and variance αh and σ2h

• Thus,

with

S(t +h) =S(t)(1+ rh)

rh =~N(αh,σ2h)


∑ =+=

n

i ihrlnSTSln

1 )1()]0(/)([

A Description of Stock Price Behavior (cont’d)

• The continuously compounded return from 0 to T is

• 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


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


Brownian Motion (cont’d)

• 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



• 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)


dttYtdZ )()( =


• Then

(20.4) where Y(t) is a random draw from a binomial

distribution, such that Y(t) is 1 with probability 50%

• Equation (20.4) 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.”


∫+=T

tdZZTZ0

)()0()(


• Since Z(T) is the sum of individual dZ(t)’s, we can write

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”


)( )( tdZdttdX σα +=

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

(20.8)

• This process is called arithmetic Brownian motion α is the instantaneous mean per unit time

σ2 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 (αT, σ2T)


∫∫ ++=TT

tdZdtXTX00

)( )0()( σα

Arithmetic Brownian Motion (cont’d)

• An integral representation of equation (20.8) is

• 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 αdt term introduces a nonrandom drift into

the process


Arithmetic Brownian Motion (cont’d)

• 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


)( )]([)( tdZdttXatdX σλ +−=

The Ornstein-Uhlenbeck Process

• We can incorporate mean reversion by modifying the drift term

(20.9)

• When α = 0, this equation is called an Ornstein-Uhlenbeck process The parameter λ measures the speed of the reversion:

If λ is large, reversion happens more quickly In the long run, we expect X to revert toward α As with arithmetic Brownian motion, X can still

become negative


)( )(

)(tdZdta

tX

tdX σ+=

)()( )( )( tdZtXdttXatdX σ+=

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)

This is an Itô process that can also be written

(20.11)

This process is known as geometric Brownian motion


∫∫ +=−TT

tdZtXdttXXTX00

)()( )( )0()( σα

Geometric Brownian Motion (cont’d)

• The percentage change in the asset value is normally distributed with instantaneous mean α and instantaneous variance σ2

• The integral representation for equation (20.11) is


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


hhtX

htX

ασ

ασ

=)()(

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

The ratio becomes infinite as h approaches dt

• As the time interval becomes longer, the mean becomes more important than the standard deviation


Relative Importance of the Drift and Noise Terms (cont’d)

• The results in the table hold for both geometric Brownian motion and arithmetic Brownian motion.


)(1)()(

)()(

22

1

1

tWtWtZ

tWtZ

' ρρ −+=

=

0 )]()([1])([)]()([ 2122

1 +=−+= ttWtWEtWEtZtZE ' ρρρ

Correlated Itô Processes

• Let W1(t) and W2(t) be independent Brownian motions. Then

(20.16) This is the Cholesky decomposition

• The correlation between Z(t) and Z'(t) is

The second term on the right-hand side is 0 because W1(t) and W2(t) are independent.


Multiplication Rules

• We can simplify complex terms containing dt and dZ by using the following “multiplication rules”

dt dZ = 0 (20.17a)

(dt)2 = 0 (20.17b)

(dZ)2 = dt (20.17c)

dZ dZ' = ρdt (20.17d) The reason behind these multiplication rules is that the

multiplications resulting in powers of dt greater than 1 vanish.


i

ii

rá

σ−

=ratio Sharpe

The Sharpe Ratio

• If asset i has expected return αi, the risk premium is defined as

Risk premiumi = αi – r where r is the risk-free rate

• The Sharpe ratio for asset i is the risk premium, αi – r, per unit of volatility, σI

(20.18)


The Sharpe Ratio (cont’d)

• 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

(20.19)

(20.20)

Because the two stock prices are driven by the same dZ, it must be the case that

dS1 =α1S1dt+σ1S1dZdS2 =α2S2dt+σ 2S2dZ

(α1 −r) / σ1 =(α2 −r) / σ 2


)()()(

)(tdZdt

tS

tdS σδα +−=

)( )()(

)( * tdZdtrtS

tdS σδ +−=

The Risk-Neutral Process

• Suppose the true price process is

where δ is the dividend yield on the stock

• We can write a risk-neutral version of this process by subtracting the risk premium, α – r, from the drift, α – δ

(20.26) The probability distribution (dZ*(t)) associated with the risk-

neutral process is said to be the risk-neutral measure When we switch to the risk-neutral process, the volatility

remains the same


Itô’s Lemma

• Suppose a stock with an expected instantaneous return of α, dividend yield of δ, and instantaneous volatility σ follows geometric Brownian motion

• 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?

dS(t) =(α −δ)S(t)dt+σS(t)dZ(t)


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

• where CS = C/S, CSS = 2C/S2, and Ct = C/t

The terms in square brackets are the expected change in the option price

dZóSCdtCCSSC

dtCdSCdSCtSdC

StSSS

tSSS

+⎥⎦

⎤⎢⎣

⎡ ++−=

++=

21

)(

)(21

) ,(

22

2

σδα


Itô’s Lemma (cont’d)

• The extra term involving the variance is the Jensen’s inequality correction due to the uncertainty of the stochastic process

If there is no uncertainty, that is if σ = 0, then Itô’s Lemma reduces to the calculation of a total derivative

dC(S, t) =CSdS+Ctdt


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

• Denote the pairwise correlations as

nidzdttS

tdSiii

i

i ., . . ,1 , )(

)(=+= σα

E(dzi ×dzj ) =ρi, jdt


∑ ∑ ∑= = =++=

n

i

n

i t

n

j SSjiiSn dtCCdSdSdSCtSSdCjii1 1 11 2

1), ., . . ,(

∑ ∑∑ = ==++=

n

i t

n

j SSjiijji

n

i Siin CCSSCStSSdCEdt jii 1 111 2

1)] , ., . . ,([

1ρσσα

Multivariate Itô’s Lemma (cont’d)

• If C(S1, . . ., Sn, t) is a twice-differentiable function of the Si’s, we have

• The expected change in C per unit time is


Valuing a Claim on Sa

• Suppose a stock with an expected instantaneous return of α, dividend yield of δ, and instantaneous volatility σ follows geometric Brownian motion

• 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

dS(t) =(α −δ)S(t)dt+σS(t)dZ(t)


TaaraarTaPT eSeTSF

])1(2

1)([

,0

2

)0(])([σδ −+−−=

TaaraaaT eSTSF

])1(2

1)([

,0

2

)0(])([σδ −+−

=

2* )1(2

1)( σδδ −−−−= aarar

Valuing a Claim on Sa (cont’d)

• Proposition 20.3:

The value at time 0 of a claim paying S(T)a—the prepaid forward price—is

(20.30)

The forward price for S(T)a is

(20.31)

• The lease rate for a claim paying Sa is


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

dtSaaS

dSaS

dtSSaadSaSdS

aa

aaa

2

221

)1(2

1

)()1(2

1

σ

σ

−+=

−+= −−


dZadtaaaS

dSa

a

σσδα +⎥⎦

⎤⎢⎣

⎡ −+−= )1(21

)( 2

The Process Followed by Sa (cont’d)

• Dividing by Sa, we get

(20.32)

Thus, Sa follows geometric Brownian motion with drift a (α – δ) + 1/2a (a – 1)σ2 and risk aσdZ

If α is the expected return for S, the expected return of a claim with price Sa will be

a (α – r) + r (20.33)

Thus, the risk premium is a (α – r)


Examples

• Claims on S Suppose a = 1. Equation (20.30) then gives us

• This is just the prepaid forward price on a stock

• 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

• This is the price of a T-period pure discount bond

TeSV δ−= )0()0(

rTeV −=)0(


TT

TTr

TrrTT

eSFeeS

eSeSF22

2

2,0

)(22

]2[22,0

)]([)0(

)0()(

σσδ

σδ

==

=−

−+−−

Examples (cont’d)

• Claims on S2

When a = 2, the claim pays S(T)2. From equation (20.30), the forward price is

(20.36)

Thus, the forward price on the squared stock price is the squared forward price times a variance term


TT

TTrT eFeeSSF

22

)]0(/1[)/1( 1,0

)(,0

σσδ −− ==

Examples (cont’d)

• Claims on 1/S If a = –1, the claim pays 1/S. Using equation

(20.31), we get

• 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.


SSSS dZdtS

dS σδα +−= )(

QQQQ dZdtQ

dQ σδα +−= )(

dtdZdZ QS ρ=

Valuing a Claim on Saqb

• Suppose a claim pays S(T)aQ(T)b, where S follows

(20.37)

and Q follows

(20.38)

where


)(,,, )()()( tTabbTt

aTt

baTt

QSeQFSFQSF −= σρσ

QSQS abba σρσσσ 22222 ++

Valuing a Claim on Saqb (cont’d)

• 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

• 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


2SQSab σσρσ =


• The squared security, S2, is a special case of Proposition 20.4

When S = Q, a = b = 1, and ρ = 1, the covariance term becomes

This gives us the same result as equation (20.36) for the forward price for a squared stock


iiiii

i dzdtS

dS σδα +−= )(

∏==

n

i

aiiStV

1)(

∏=

∑ ∑=−

= +=n

i

aaaiTT

n

i

n

ij jijiiji eSFVF

1

,0,0

1

1 1)]([)(σσρ


• Proposition 20.4 can be generalized Suppose there are n stocks, each of which follows the

process

(20.41)

where dzidzj = ρijdt. Let

(20.42) The forward price for V is then

(20.43)


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.


Jumps in the Stock Price (cont’d)

• 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 λ is the expected number of jumps per unit time over an interval dt, then

Prob(jump) = λdt

Prob(no jump) = 1 – λdt



• We can write the stock price process as

(20.44)

where

0 if there is no jump

dq =

Y-1 if there is a jump

and E(dq) = λkdt When no jump is occurring, the stock price S evolves as

geometric Brownian motion. When the jump occurs, the new stock price is YS.

dS(t) / S(t) =(α −λk)dt+σdZ+dq


)] ,() ,([2

1) ,( 22 tSCtSYCEdtCdtSCdSCtSdC YtSSS −+++= λσ


• Proposition 20.5

Suppose an asset follows equation (20.44). If C(S, t) is a twice continuously differentiable function of the stock price, the process followed by C is

(20.45)

• The last term in the equation is the expected change in the option price conditional on the jump times the probability of the jump.

Documents

Chapter 20 Brownian Motion and Itô’s Lemma. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 20-2 Introduction Stock and other asset prices