This paper is set to analytically describe properties of the
hyperbolic distribution. This law, along with the variance-gamma
distribution, is one of the most popular normal mean–variance
mixtures from the point of view of various applications. We have
found closed form expressions for the cumulative distribution and
partial moment-generating functions of the hyperbolic distribution.
The obtained formulas use the values of the Humbert confluent
hypergeometric and Whittaker special functions. The results are
applied to the problem of European option pricing in the related
Levy model of financial market. The research demonstrates that the
discussed normal mean–variance mixture is analytically tractable.