Chapter 2: FAQs 209
Why is the Lognormal Distribution
Important?
Short Answer
The lognormal distribution is often used as a model
for the distribution of equity or commodity prices,
exchange rates and indices. Thenormaldistribution
is often used to modelreturns.
Example
The stochastic differential equation commonly used to
represent stocks,
dS=μSdt+σSdX
results in a lognormal distribution forS,providedμand
σare not dependent on stock price.
Long Answer
A quantity is lognormally distributed if its logarithm is
normally distributed, that is the definition of lognormal.
The probability density function is
1
√
2 πbx
exp
(
−
1
2 b^2
(ln(x)−a)^2
)
x≥0,
where the parametersaandb>0 represent location
and scale. The distribution is skewed to the right,
extending to infinity and bounded below by zero. (The
left limit can be shifted to give an extra parameter, and
it can be reflected in the vertical axis so as to extend to
minus infinity instead.)
If we have the stochastic differential equation above
then the probability density function forSin terms of
time and the parameters is
1
σS
√
2 πt
e
−
(
ln(S/S 0 )−(μ−^12 σ^2 )t
) 2
/ 2 σ^2 t
,
whereS 0 is the value ofSat timet=0.
