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On Properties of the Hyperbolic Distribution

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  • Roman V. Ivanov

    (Laboratory of Control under Incomplete Information, V.A. Trapeznikov Institute of Control Sciences of RAS, Profsoyuznaya 65, 117997 Moscow, Russia)

Abstract

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 Lévy model of financial market. The research demonstrates that the discussed normal mean–variance mixture is analytically tractable.

Suggested Citation

  • Roman V. Ivanov, 2024. "On Properties of the Hyperbolic Distribution," Mathematics, MDPI, vol. 12(18), pages 1-20, September.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:18:p:2888-:d:1479168
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    References listed on IDEAS

    as
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    7. Roman V. Ivanov & Katsunori Ano, 2016. "On exact pricing of FX options in multivariate time-changed Lévy models," Review of Derivatives Research, Springer, vol. 19(3), pages 201-216, October.
    8. Helyette Geman & C. Peter M. Dilip Y. Marc, 2007. "Self decomposability and option pricing," Post-Print halshs-00144193, HAL.
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