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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 )   
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2
e
2
dx
18-3
The Normal Distribution (cont’d)
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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
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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)
x3
~ N (0,1) , and 3  5  z ~ N (3,25)
then
5
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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 = rijij
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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
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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

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18-9
The Lognormal Distribution (cont’d)
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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)] = nh2
• If returns are iid, the mean and variance of the
continuously compounded returns are proportional
to time
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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 (d2 )

where the expression contains a, the true expected return on
the stock in place of r, the risk-free rate
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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
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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 (d1 )
E ( St | St  K )  Se
N (d2 )
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 )
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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
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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
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18-16
How Are Asset Prices Distributed?
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18-17
How Are Asset Prices Distributed? (cont’d)
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18-18
How Are Asset Prices Distributed? (cont’d)
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18-19