Quantitative Methods: Application

Reading 9. Common Probability Distributions

Learning Outcome Statements

o. explain the relationship between normal and lognormal distributions and why the lognormal distribution is used to model asset prices;

CFA Curriculum, 2020, Volume 1

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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.

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User Contributed Comments 6

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rufi

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bahodir

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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