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In probability theory, the lognormal distribution is defined with reference to
the normal distribution. A random variable X
is
lognormal if its
natural logarithm,
.gif){kind=small}
,
is normal.
Consider a stochastic process
representing accumulated values over time for some asset. It might
represent daily accumulated values of a common stock or a physical
commodity position. At some time t, the realization
of
is
known, but the realization of the subsequent value
is unknown. Accordingly, the single-period log return
is random. Assume that its conditional
as of time t is
normal. Then, by the above definition,
must be lognormal, and it is easy to demonstrate (try it!) that
is also lognormal. This model—of log returns being normal and
corresponding prices being lognormal—is one of the most ubiquitous models
in finance. It is the reason that the lognormal
plays such an
important role in finance.
The lognormal
for a random variable X
may be specified with its mean μ and
variance
.
Alternatively, it may be specified with the mean m and variance
s2 of the normally distributed log(X). We
denote a lognormal
as
,
but its probability density function (PDF) is most easily expressed in
terms of m and s:
 |
[1] |
This is graphed in Exhibit 1:
|
λ(μ,σ2) Probability Density Function |
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The λ(μ,σ2) distribution is positively skewed. |
The expectation, standard deviation,
skewness and
kurtosis of a lognormal
are, in terms of m and s:
 |
[2] |
 |
[3] |
 |
[4] |
 |
[5] |
If we know μ and σ instead of m and s, we can convert between these with:
 |
[6] |
 |
[7] |
The reverse conversion is provided by [2] and [3].
As with the normal
, the cumulative
function (CDF) of a lognormal
exists but cannot
be expressed in terms of standard functions. Values can be inferred from
appropriate values of the standard normal CDF.
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