Chapter 20 Brownian Motion and Itô’s Lemma

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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   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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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) e R(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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved A Description of Stock Price Behavior (cont’d) Suppose that  h, the length of each time period, is small,  the return on the stock, r h, is normally distributed with mean and variance  h and  2 h Thus, with

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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 + s 1 ) – Z(t) is independent of Z(t) – Z(t – s 2 ), where s 1, s 2 >0. That is, nonoverlapping increments are independently distributed  Z(t) is continuous

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Brownian Motion (cont’d) 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)

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Brownian Motion (cont’d) 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.”

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Brownian Motion (cont’d) 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”

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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   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,  2 T)

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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  dt term introduces a nonrandom drift into the process

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved The Ornstein-Uhlenbeck Process We can incorporate mean reversion by modifying the drift term (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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Geometric Brownian Motion (cont’d) 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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

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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Correlated Itô Processes Let W 1 (t) and W 2 (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 W 1 (t) and W 2 (t) are independent.

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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' =  dt (20.17d)  The reason behind these multiplication rules is that the multiplications resulting in powers of dt greater than 1 vanish.

Copyright © 2006 Pearson Addison-Wesley. All rights reserved The Sharpe Ratio If asset i has expected return  i, the risk premium is defined as Risk premium i =  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 (20.18)

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved The Risk-Neutral Process Suppose the true price process is  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,  –  (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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Itô’s Lemma Suppose a stock with an expected instantaneous return of , dividend yield of , and instantaneous volatility  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?

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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 C S =  C/  S, C SS =  2 C/  S 2, and C t =  C/  t  The terms in square brackets are the expected change in the option price

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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  = 0, then Itô’s Lemma reduces to the calculation of a total derivative

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Multivariate Itô’s Lemma (cont’d) If C(S 1,..., S n, t) is a twice-differentiable function of the S i ’s, we have The expected change in C per unit time is

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Valuing a Claim on S a Suppose a stock with an expected instantaneous return of , dividend yield of , and instantaneous volatility  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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Valuing a Claim on S a (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 S a is

Copyright © 2006 Pearson Addison-Wesley. All rights reserved The Process Followed by S a 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 S a

Copyright © 2006 Pearson Addison-Wesley. All rights reserved The Process Followed by S a (cont’d) Dividing by S a, we get (20.32)  Thus, S a 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 S a will be a (  – r) + r (20.33)  Thus, the risk premium is a (  – r)

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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 S 0  If a = 0, the claim does not depend on the stock price. Since S 0 = 1, it is a bond. Setting a = 0 gives us This is the price of a T-period pure discount bond

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Examples (cont’d) Claims on S 2  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

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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Valuing a Claim on S a q b Suppose a claim pays S(T) a Q(T) b, where S follows (20.37) and Q follows (20.38) where

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Valuing a Claim on S a q b (cont’d) Proposition 20.4  Suppose that the forward prices for S a and Q b are given by Proposition Then the forward price for S a Q b is The forward price for S a Q b is the product of those two forward prices times a factor that accounts for the covariance between the two assets  The variance of S a Q b is given by

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Valuing a Claim on S a q b (cont’d) The squared security, S 2, is a special case of Proposition 20.4  When S = Q, a = b = 1, and  = 1, the covariance term becomes  This gives us the same result as equation (20.36) for the forward price for a squared stock

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Valuing a Claim on S a q b (cont’d) Proposition 20.4 can be generalized  Suppose there are n stocks, each of which follows the process (20.41) where dz i dz j =  ij dt. Let (20.42)  The forward price for V is then (20.43)

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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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 is the expected number of jumps per unit time over an interval dt, then Prob(jump) = dt Prob(no jump) = 1 – dt

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Jumps in the Stock Price (cont’d) 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) = 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.

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Jumps in the Stock Price (cont’d) 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.