Generalized Hyperbolic Distribution

μ

0

λ

-0.5

δ

0.5

α

1

β

0.5

mean: 0.288675

variance: 0.7698

skewness: 2.27951

excess kurtosis: 13.8564

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

μ

and

δ

describe the location and the scale, while

α

and

β

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

λ

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

λ=1

gives the hyperbolic distribution and

λ=-1/2

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

δ=0

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).