Chapter 18 The Lognormal Distribution

Copyright © 2006 Pearson Addison-Wesley. All rights reserved The Normal Distribution Normal distribution (or density)

Copyright © 2006 Pearson Addison-Wesley. All rights reserved The Normal Distribution (cont’d) Normal density is symmetric: If a random variable x is normally distributed with mean  and standard deviation,  z is a random variable distributed standard normal: The value of the cumulative normal distribution function N(a) equals to the probability P of a number z drawn from the normal distribution to be less than a. [P(z<a)]

Copyright © 2006 Pearson Addison-Wesley. All rights reserved The Normal Distribution (cont’d)

Copyright © 2006 Pearson Addison-Wesley. All rights reserved The Normal Distribution (cont’d) The probability of a number drawn from the standard normal distribution will be between a and –a: Prob (z < –a) = N(–a) Prob (z < a) = N(a) therefore Prob (–a < z < a) = N(a) – N(–a) = N(a) – [1 – N(a)] = 2·N(a) – 1 Example: Prob (–0.3 < z < 0.3) = 2· – 1 =

Copyright © 2006 Pearson Addison-Wesley. All rights reserved The Normal Distribution (cont’d) Converting a normal random variable to standard normal:  If, then if And vice versa:  If, then if Example 18.2: Suppose and then, and

Copyright © 2006 Pearson Addison-Wesley. All rights reserved The Normal Distribution (cont’d) The sum of normal random variables is also  where x i, i = 1,…,n, are n random variables, with mean E(x i ) =  i, variance Var(x i ) =  i 2, covariance Cov(x i,x j ) =  ij =  ij  i  j

Copyright © 2006 Pearson Addison-Wesley. All rights reserved The Lognormal Distribution A random variable x is lognormally distributed if ln(x) is normally distributed  If x is normal, and ln(y) = x (or y = e x ), then y is lognormal  If continuously compounded stock returns are normal then the stock price is lognormally distributed Product of lognormal variables is lognormal  If x 1 and x 2 are normal, then y 1 =e x 1 and y 2 =e x 2 are lognormal  The product of y 1 and y 2 : y 1 x y 2 = e x 1 x e x 2 = e x 1 +x 2  Since x 1 +x 2 is normal, e x 1 +x 2 is lognormal

Copyright © 2006 Pearson Addison-Wesley. All rights reserved The Lognormal Distribution (cont’d) The lognormal density function  where S 0 is initial stock price, and ln(S/S 0 )~N(m,v 2 ), S is future stock price, m is mean, and v is standard deviation of continuously compounded return If x ~ N(m,v 2 ), then

Copyright © 2006 Pearson Addison-Wesley. All rights reserved The Lognormal Distribution (cont’d)

Copyright © 2006 Pearson Addison-Wesley. All rights reserved A Lognormal Model of Stock Prices If the stock price S t is lognormal, S t / S 0 = e x, where x, the continuously compounded return from 0 to t is normal If R(t, s) is the continuously compounded return from t to s, and, t 0 < t 1 < t 2, then R(t 0, t 2 ) = R(t 0, t 1 ) + R(t 1, t 2 ) From 0 to T, E[R(0,T)] = n  h, and Var[R(0,T)] = n  h 2 If returns are iid, the mean and variance of the continuously compounded returns are proportional to time

Copyright © 2006 Pearson Addison-Wesley. All rights reserved A Lognormal Model of Stock Prices (cont’d) If we assume that then and therefore If current stock price is S 0, the probability that the option will expire in the money, i.e.  where the expression contains , the true expected return on the stock in place of r, the risk-free rate

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Lognormal Probability Calculations Prices S t L and S t U such that Prob (S t L S t ) = p/2

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Lognormal Probability Calculations (cont’d) Given the option expires in the money, what is the expected stock price? The conditional expected price  where the expression contains a, the true expected return on the stock in place of r, the risk-free rate The Black-Scholes formula—the price of a call option on a nondividend-paying stock

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Estimating the Parameters of a Lognormal Distribution The lognormality assumption has two implications  Over any time horizon continuously compounded return is normal  The mean and variance of returns grow proportionally with time

Copyright © 2006 Pearson Addison-Wesley. All rights reserved Estimating the Parameters of a Lognormal Distribution (cont’d) The mean of the second column is and the standard deviation is Annualized standard deviation Annualized expected return

Copyright © 2006 Pearson Addison-Wesley. All rights reserved How Are Asset Prices Distributed?

Copyright © 2006 Pearson Addison-Wesley. All rights reserved How Are Asset Prices Distributed? (cont’d)

Copyright © 2006 Pearson Addison-Wesley. All rights reserved How Are Asset Prices Distributed? (cont’d)