The Most Popular Probability Distributions and Their Uses in Finance

Random variables can be continuous or discrete (the latter denoted below by *). Or a combination. New distributions can also be made up using random variables from two or more distributions.

Here is a list of distributions seen in finance (mostly), and some words on each.

Normal or Gaussian

This distribution is unbounded below and above, and is symmetrical about its mean. It has two parameters: a, location; b > 0 scale. Its probability density function is given by

This distribution is commonly used to model equity returns, and, indeed, the changes in many financial quantities. Errors in observations of real phenomena are often normally distributed. The normal distribution is also common because of the Central Limit Theorem.

Mean a.

Variance b2.

Lognormal

Bounded below, unbounded above. It has two parameters: a, location; b > 0 scale. Its probability density function is given by

This distribution is commonly used to model equity prices. Lognormality of prices follows from the assumption of normally distributed returns.

Mean

Variance

Poisson*

The random variables take non-negative integer values only. The distribution has one parameter: a > 0. Its probability density function is given by

This distribution is used in credit risk modelling, representing the number of credit events in a given time.

Mean a.

Variance a.

Chi square

Bounded below and unbounded above. It has two parameters a > 0, the location; v, an integer, the degrees of freedom. Its probability density function is given by

where T() is the Gamma function. The chi-square distribution comes from adding up the squares of v normally distributed random variables. The chi-square distribution with one degree of freedom is the distribution of the hedging error from an option that is hedged only discretely. It is therefore a very important distribution in option practice, if not option theory.

Mean v + a.

Variance 2(v + 2a).

Gumbel

Unbounded above and below. It has two parameters: a, location; b > 0 scale. Its probability density function is given by

The Gumbel distribution is useful for modelling extreme values, representing the distribution of the maximum value out of a large number of random variables drawn from an unbounded distribution.

Mean a + y b,

where y is Euler's constant, 0.577216....

Variance

Weibull

Bounded below and unbounded above. It has three parameters: a, location; b > 0, scale; c > 0, shape. Its probability density function is given by

The Weibull distribution is also useful for modelling extreme values, representing the distribution of the maximum value out of a large number of random variables drawn from a bounded distribution. (The figure shows a 'humped' Weibull, but depending on parameter values the distribution can be monotonie.)

Mean

Variance

where r() is the Gamma function.

Student's t

Unbounded above and below. It has three parameters: a, location; b > 0, scale; c > 0, degrees of freedom. Its probability density function is given by

where T() is the Gamma function. This distribution represents small-sample drawings from a normal distribution. It is also used for representing equity returns.

Mean a.

Variance

Note that the nth moment only exists if c > n.

Pareto

Bounded below, unbounded above. It has two parameters: a > 0, scale; b > 0 shape. Its probability density function is given by

Commonly used to describe the distribution of wealth, this is the classical power-law distribution.

Mean

Variance

Note that the nth moment only exists if b > n.

Uniform

Bounded below and above. It has two location parameters, a and b. Its probability density function is given

by

Mean

Variance

Inverse normal

Bounded below, unbounded above. It has two parameters: a > 0, location; b > 0 scale. Its probability density function is given by

This distribution models the time taken by a Brownian motion to cover a certain distance.

Mean a.

Variance

Gamma

Bounded below, unbounded above. It has three parameters: a, location; b > 0 scale; c > 0 shape. Its probability density function is given by

where r() is the Gamma function. When c = 1 this is the exponential distribution and when a = O and b = 2 this is the chi-square distribution with 2c degrees of freedom.

Mean a + bc.

Variance b2 c.

Logistic

This distribution is unbounded below and above. It has two parameters: a, location; b > 0 scale. Its probability density function is given by

The logistic distribution models the mid value of highs and lows of a collection of random variables, as the number of samples becomes large.

Mean a.

Variance

Laplace

This distribution is unbounded below and above. It has two parameters: a, location; b > 0 scale. Its probability

density function is given by

Errors in observations are usually either normal or Laplace. Mean a.

Variance 2b2.

Cauchy

This distribution is unbounded below and above. It has two parameters: a, location; b > 0 scale. Its probability density function is given by

This distribution is rarely used in finance. It does not have any finite moments, but its mode and median are both a.

Beta

This distribution is bounded below and above. It has four parameters: a, location of lower limit; b > a location of upper limit; c > 0and d > 0 shape. Its probability density function is given by

where r() is the Gamma function. This distribution is rarely used in finance.

Mean

Variance

Exponential

Bounded below, unbounded above. It has two parameters: a, location; b > 0 scale. Its probability density function is given by

This distribution is rarely used in finance. Mean a + b.

Variance b2.

Levy Unbounded below and above. It has four parameters: X, a location (mean); 0 <a < 2, the peakedness; —1 </3 < 1, the skewness; v > 0, a spread. (Conventional notation is used here.) This distribution has been saved to last because its probability density function does not have a simple closed form. Instead it must be written in terms of its characteristic function. If P(x) is the probability density function then the

moment generating function is given by

where i = V^T. For the Levy distribution

The normal distribution is a special case of this with a = 2 and { = 0, and with the parameter v being one half of the variance. The Levy distribution, or Pareto Levy distribution, is increasingly popular in finance because it matches data well, and has suitable fat tails. It also has the important theoretical property of being a stable distribution in that the sum of independent random numbers drawn from the Levy distribution will itself be Levy. This is a useful property for the distribution of returns. If you add up n independent numbers from the Levy distribution with the above parameters then you will get a number from another Levy distribution with the same a and f but with mean of nl/ax and spread nl/av. The tail of the distribution decays like |x|—1—a.

Mean i.

Variance infinite, unless a = 2, when it is 2v.

References and Further Reading

Spiegel, MR, Schiller & JJ Srinivasan, RA 2000 Schaum's Outline of Probability and Statistics. McGraw-Hill

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