Subject 10. The Lognormal Distribution

The key properties of the normal distribution have been presented in the LOS above. As a summary, some of the key properties of this distribution are:

  • It is symmetrical about the mean.
  • It has zero skewness.
  • It has a kurtosis of 3.

A random variable, Y, follows a lognormal distribution if its natural logarithm, lnY, is normally distributed. You can think of the term lognormal as "the log is normal." For example, suppose X is a normal random variable, and Y = eX. Therefore, LnY = Ln(eX) = X. Because X is normally distributed, Y follows a lognormal distribution.

  • Like the normal distribution, the lognormal distribution is completely described by two parameters: mean and variance.
  • Unlike the normal distribution, the lognormal distribution is defined in terms of the parameters of the associated normal distribution. Note that the mean of Y is not equal to the mean of X, and the variance of Y is not equal to the variance of X. In contrast, the normal distribution is defined by its own mean and variance.
  • The lognormal distribution is bounded below by 0. In contrast, the normal distribution extends to negative infinity without limit.
  • The lognormal distribution is skewed to the right (i.e., it has a long right tail). In contrast, the normal distribution is bell-shaped (i.e., it is symmetrical).

The reverse is also true; if a random variable Y follows a lognormal distribution, then its natural logarithm, lnY, is normally distributed.

User Contributed Comments 6

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rufi: this is a good tool for BSOPM
bahodir: what is BSOPM?
bobert: Black-Scholes Option Pricing Model
Seemorr: What kind of variable would be lognormally distributed, but not normally?
riouxcf: Some variables which have frequent outliers can be made more normal by taking the log. The normal distribution tends to underestimate extremes.
czar: Seemor: stock prices (log) and stock returns (normal) as stock prices lowest value can only be 0 while returns can be negative