Lognormal Distribution

Explained: lognormal distribution

In probability theory, the lognormal distribution is defined with reference to the normal distribution. A random variable X is lognormal if its natural logarithm, , 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 Exhibit 1

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.

Related Internal Links
Cauchy distribution A bell-shaped distribution that is more peaked and has fatter tails than the normal distribution. chi-squared distribution If you square a normal random variable, the result is a chi-squared random variable. expected value A parameter describing the "center of gravity" of a distribution. joint normal distribution A multivariate distribution with normal marginal distributions. kurtosis A parameter describing the peakedness and tails of a distribution. normal distribution Perhaps the most important probability distribution for probability and statistics. quantile A notion from probability that can be used as a parameter. skewness A parameter that describes the lack of symmetry of a distribution. standard deviation A parameter describing the dispersion of a distribution. uniform distribution A continuous probability distribution that has constant probability on a finite interval.
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Evans, Hastings and Peacock (2000) is a handy reference with detailed information on numerous probability distributions. Statistical Distributions M. Evans, N. Hastings, B. Peacock quality   technical   2000
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