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Chapter 18 The Lognormal Distribution The Normal Distribution • Normal distribution (or density) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 1 ( x; , ) e 2 1 x 2 2 18-2 The Normal Distribution (cont’d) • Normal density is symmetric: ( x; , ) ( x; , ) • If a random variable x is normally distributed with mean and standard deviation, x ~ N ( , 2 ) • z is a random variable distributed standard normal: z ~ N (0,1) • 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 1 2 be less than a. [P(z<a)] x a 1 N (a ) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 2 e 2 dx 18-3 The Normal Distribution (cont’d) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 18-4 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) = 2·0.6179 – 1 = 0.2358 Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 18-5 The Normal Distribution (cont’d) • Converting a normal random variable to standard normal: If x ~ N ( , 2 ) , then z ~ N (0,1) if • And vice versa: z x If z ~ N (0,1) , then x ~ N ( , 2 ) if x z • Example 18.2: Suppose x ~ N (3,5) and z ~ N (0,1) x3 ~ N (0,1) , and 3 5 z ~ N (3,25) then 5 Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 18-6 The Normal Distribution (cont’d) • The sum of normal random variables is also n n n i xi ~ N i i , i j ij i 1 i 1 i 1 j 1 n where xi, i = 1,…,n, are n random variables, with mean E(xi) = i, variance Var(xi) =i2, covariance Cov(xi,xj) = ij = rijij Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 18-7 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 = ex), 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 x1 and x2 are normal, then y1=ex1 and y2=ex2 are lognormal The product of y1 and y2: y1 x y2 = ex1 x ex2 = ex1+x2 Since x1+x2 is normal, ex1+x2 is lognormal Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 18-8 The Lognormal Distribution (cont’d) • The lognormal density function 1 g ( S ; m, v , S0 ) e Sv 2 1 ln( S ) [ln( S 0 ) m 0.5v 2 ] 2 v 2 where S0 is initial stock price, and ln(S/S0)~N(m,v2), S is future stock price, m is mean, and v is standard deviation of continuously compounded return 1 2 2 • If x ~ N(m,v ), then m v x E (e ) e 2 Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 18-9 The Lognormal Distribution (cont’d) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 18-10 A Lognormal Model of Stock Prices • If the stock price St is lognormal, St / S0 = ex, 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, t0 < t1 < t2, then R(t0, t2) = R(t0, t1) + R(t1, t2) • From 0 to T, E[R(0,T)] = nah , and Var[R(0,T)] = nh2 • 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. 18-11 A Lognormal Model of Stock Prices (cont’d) • If we assume that In(St / S0 ) ~ N[(a 0.5 2 )t, 2t] then In(St / S0 ) (a 0.5 2 )t t z and therefore St S0e (a 0.5 2 ) t t z • If current stock price is S0, the probability that the option will expire in the money, i.e. Prob( St K ) N (d2 ) where the expression contains a, the true expected return on the stock in place of r, the risk-free rate Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 18-12 Lognormal Probability Calculations • Prices StL and StU such that Prob (StL < St ) = p/2 and Prob (StU > St ) = p/2 StL S0 1 ( a 2 ) t t N 1 ( p / 2 ) e 2 Copyright © 2006 Pearson Addison-Wesley. All rights reserved. StU S0 1 ( a 2 ) t t N 1 ( p / 2 ) e 2 18-13 Lognormal Probability Calculations (cont’d) • Given the option expires in the money, what is the expected stock price? The conditional expected price N (d1 ) E ( St | St K ) Se N (d2 ) where the expression contains a, the true expected return on the stock in place of r, the risk-free rate (a )t • The Black-Scholes formula—the price of a call option on a nondividend-paying stock P( S , K , , r , t , ) Ke rt N ( d 2 ) e t SN ( d1 ) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 18-14 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. 18-15 Estimating the Parameters of a Lognormal Distribution (cont’d) • The mean of the second column is 0.006745 and the standard deviation is 0.038208 • Annualized standard deviation 0.038208 52 0.2755 • Annualized expected return 0.006745 52 0.5 0.2755 0.2755 0.3877 Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 18-16 How Are Asset Prices Distributed? Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 18-17 How Are Asset Prices Distributed? (cont’d) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 18-18 How Are Asset Prices Distributed? (cont’d) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 18-19