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The Semi-Hyperbolic Distribution and Its Applications

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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 studies a subclass of the class of generalized hyperbolic distribution called the semi-hyperbolic distribution. We obtain analytical expressions for the cumulative distribution function and, specifically, their first and second lower partial moments. Using the received formulas, we compute the value at risk, the expected shortfall, and the semivariance in the semi-hyperbolic model of the financial market. The formulas depend on the values of generalized hypergeometric functions and modified Bessel functions of the second kind. The research illustrates the possibility of analysis of generalized hyperbolic models using the same methodology as is employed for the well-established variance-gamma model.

Suggested Citation

  • Roman V. Ivanov, 2023. "The Semi-Hyperbolic Distribution and Its Applications," Stats, MDPI, vol. 6(4), pages 1-21, October.
    • Handle: RePEc:gam:jstats:v:6:y:2023:i:4:p:71-1146:d:1264669
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    References listed on IDEAS

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    1. Dilip B. Madan & Peter P. Carr & Eric C. Chang, 1998. "The Variance Gamma Process and Option Pricing," Review of Finance, European Finance Association, vol. 2(1), pages 79-105.
    2. Cai, Zongwu & Wang, Xian, 2008. "Nonparametric estimation of conditional VaR and expected shortfall," Journal of Econometrics, Elsevier, vol. 147(1), pages 120-130, November.
    3. Yannick Armenti & Stéphane Crépey & Samuel Drapeau & Antonis Papapantoleon, 2018. "Multivariate Shortfall Risk Allocation and Systemic Risk," Working Papers hal-01764398, HAL.
    4. Athanassios N. Avramidis & Pierre L'Ecuyer, 2006. "Efficient Monte Carlo and Quasi-Monte Carlo Option Pricing Under the Variance Gamma Model," Management Science, INFORMS, vol. 52(12), pages 1930-1944, December.
    5. Roman V. Ivanov, 2018. "A Credit-Risk Valuation under the Variance-Gamma Asset Return," Risks, MDPI, vol. 6(2), pages 1-25, May.
    6. Eberlein, Ernst & Keller, Ulrich & Prause, Karsten, 1998. "New Insights into Smile, Mispricing, and Value at Risk: The Hyperbolic Model," The Journal of Business, University of Chicago Press, vol. 71(3), pages 371-405, July.
    7. Martin Wallmeier & Martin Diethelm, 2012. "Multivariate downside risk: Normal versus Variance Gamma," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 32(5), pages 431-458, May.
    8. Mafusalov, Alexander & Uryasev, Stan, 2016. "CVaR (superquantile) norm: Stochastic case," European Journal of Operational Research, Elsevier, vol. 249(1), pages 200-208.
    9. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-524, October.
    10. Roman V. Ivanov, 2022. "The Risk Measurement under the Variance-Gamma Process with Drift Switching," JRFM, MDPI, vol. 15(1), pages 1-27, January.
    11. So Yeon Chun & Alexander Shapiro & Stan Uryasev, 2012. "Conditional Value-at-Risk and Average Value-at-Risk: Estimation and Asymptotics," Operations Research, INFORMS, vol. 60(4), pages 739-756, August.
    12. David Scott & Diethelm Würtz & Christine Dong & Thanh Tran, 2011. "Moments of the generalized hyperbolic distribution," Computational Statistics, Springer, vol. 26(3), pages 459-476, September.
    13. Xiangyu Han & Chuancun Yin, 2022. "Tail Conditional Moments for Location-Scale Mixture of Elliptical Distributions," Mathematics, MDPI, vol. 10(4), pages 1-21, February.
    14. Martin Schweizer & Danijel Zivoi & Mario Šikić, 2018. "Dynamic Mean–Variance Optimization Problems With Deterministic Information," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(02), pages 1-38, March.
    15. Janine Balter & Alexander J. McNeil, 2018. "On the Basel Liquidity Formula for Elliptical Distributions," Papers 1803.07590, arXiv.org.
    16. J. Baixauli & Susana Alvarez, 2006. "Evaluating effects of excess kurtosis on VaR estimates: Evidence for international stock indices," Review of Quantitative Finance and Accounting, Springer, vol. 27(1), pages 27-46, August.
    17. Fred Espen Benth & Luca Di Persio & Silvia Lavagnini, 2018. "Stochastic Modeling of Wind Derivatives in Energy Markets," Risks, MDPI, vol. 6(2), pages 1-21, May.
    18. Bauer, Christian, 2000. "Value at risk using hyperbolic distributions," Journal of Economics and Business, Elsevier, vol. 52(5), pages 455-467.
    19. Robert Jarrow & Feng Zhao, 2006. "Downside Loss Aversion and Portfolio Management," Management Science, INFORMS, vol. 52(4), pages 558-566, April.
    20. Mei-Ling Ting Lee & George A. Whitmore, 2022. "Multivariate Threshold Regression Models with Cure Rates: Identification and Estimation in the Presence of the Esscher Property," Stats, MDPI, vol. 5(1), pages 1-18, February.
    21. Stefan Stöckl & Andreas Rathgeber & J. Stadler, 2017. "Fitting Generalized Hyperbolic processes - new insights for generating initial values," Post-Print hal-01768898, HAL.
    22. Elton A. Daal & Dilip B. Madan, 2005. "An Empirical Examination of the Variance-Gamma Model for Foreign Currency Options," The Journal of Business, University of Chicago Press, vol. 78(6), pages 2121-2152, November.
    23. Ole Eiler Barndorff‐Nielsen & Robert Stelzer, 2005. "Absolute Moments of Generalized Hyperbolic Distributions and Approximate Scaling of Normal Inverse Gaussian Lévy Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(4), pages 617-637, December.
    24. Martin Schweizer & Danijel Zivoi & Mario Sikic, 2017. "Dynamic Mean-Variance Optimisation Problems with Deterministic Information," Swiss Finance Institute Research Paper Series 17-29, Swiss Finance Institute, revised Feb 2018.
    25. Janine Balter & Alexander J. McNeil, 2018. "On the Basel Liquidity Formula for Elliptical Distributions," Risks, MDPI, vol. 6(3), pages 1-13, September.
    26. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    27. 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.
    28. Rockafellar, R. Tyrrell & Uryasev, Stanislav, 2002. "Conditional value-at-risk for general loss distributions," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1443-1471, July.
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