Chapter 18 The Lognormal Distribution. Copyright © 2006 Pearson Addison-Wesley. All rights...

transcript

Chapter 18

The Lognormal Distribution

( ; , )x e

The Normal Distribution

• Normal distribution (or density)

),;(),;( xx

x N~ ( , ) 2

z N~ ( , )0 1

N a e dxxa

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)]

• 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

• Converting a normal random variable to standard normal: If , then if

• And vice versa: If , then if

• Example 18.2: Suppose and then , and

z N~ ( , )0 1x N~ ( , ) 2 zx

x N~ ( , ) 2z N~ ( , )0 1 x z

x N~ ( , )3 5 z N~ ( , )0 1x

0 1~ ( , ) 3 5 3 25 z N~ ( , )

• The sum of normal random variables is also

where xi, i = 1,…,n, are n random variables, with mean E(xi) = i, variance Var(xi) =i

2, covariance Cov(xi,xj) = ij = ijij

i i i i i j ijj

nx N~ ,

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=ex

2 are lognormal

The product of y1 and y2: y1 x y2 = ex1 x ex

2 = ex1+x

Since x1+x2 is normal, ex1+x

2 is lognormal

The Lognormal Distribution (cont’d)

• The lognormal density function

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

• If x ~ N(m,v2), then

g S m v SSv

S S m v

v( ; , , )

ln( ) [ln( ) . ]

E e ex m v( )

The Lognormal Distribution (cont’d)

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)] = nh , and Var[R(0,T)] = nh2

• If returns are iid, the mean and variance of the continuously compounded returns are proportional to time

Prob( ) ( )S K N dt 2

A Lognormal Model of Stock Prices (cont’d)

• If we assume that

and therefore

• If current stock price is S0, 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

2( 0.5 )0

t t ztS S e

In(St / S0 ) ~ N[( 0.5 2 )t, 2t]2

0( / ) ( 0.5 )tIn S S t t z

S S etL t t N p

222 1( ) ( / )

S S etU t t N p

222 1( ) ( / )

Lognormal Probability Calculations

• Prices StL and St

U such that Prob (StL < St ) = p/2 and

Prob (StU > St ) = p/2

E S S K SeN d

N dt t

t( | )( )( )

P S K r t Ke N d e SN drt t( , , , , , ) ( ) ( ) 2 1

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

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

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

• Annualized expected return

0.038208 52 0.2755

0.00674552 0.50.27550.2755 0.3877

How Are Asset Prices Distributed?

How Are Asset Prices Distributed? (cont’d)

Chapter 18 The Lognormal Distribution. Copyright © 2006 Pearson Addison-Wesley. All rights...

Documents