Generalized Hyperbolic Distribution

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).


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Generalized hyperbolic distributions were introduced by Barndorff–Nielsen [1]. They generalize many previously known distributions in addition to being a source of many new ones. As they are infinitely divisible, Lévy and other stochastic processes can be based on them.
Such processes were first applied in finance by Eberlein and Keller [2]. Being normal variance-mean mixtures, GH distributions possess semi-heavy tails and allow for a natural definition of volatility models by replacing the mixing generalized the inverse Gaussian (GIG) distribution by appropriate volatility processes.
[1] O. E. Barndorff-Nielsen, "Exponentially Decreasing Distributions for the Logarithm of Particle Size," Proceedings of the Royal Society London A, 353(1674), 1977 pp. 401–419.
[2] E. Eberlein and U. Keller, "Hyperbolic Distributions in Finance," Bernoulli 1(3), 1995 pp. 281–299.
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