This Demonstration shows the probability density function of the generalized hyperbolic distribution, which generalizes a large number of distributions with numerous applications in finance and other areas. There are a number of parametrizations of this distribution. We use the most common one with five parameters, of which

and

describe the location and the scale, while

and

determine the shape of the distribution. Special values of the parameter

often correspond to well-known special cases: for example,

gives the hyperbolic distribution and

gives the normal inverse Gaussian (NIG) distribution. On the other hand, the case

gives the variance gamma distribution. Many other distributions are obtained as limiting cases.

This Demonstration also shows the effect of varying different parameters on the mean, variance, skewness, and the excess kurtosis of the distribution. (These effects are usually clearer when other parametrizations are used).