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Bilateral gamma distributions and processes in financial mathematics

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  • Küchler, Uwe
  • Tappe, Stefan

Abstract

We present a class of Lévy processes for modelling financial market fluctuations: bilateral Gamma processes. Our starting point is to explore the properties of bilateral Gamma distributions, and then we turn to their associated Lévy processes. We treat exponential Lévy stock models with an underlying bilateral Gamma process as well as term structure models driven by bilateral Gamma processes, and apply our results to a set of real financial data (DAX 1996-1998).

Suggested Citation

  • Küchler, Uwe & Tappe, Stefan, 2008. "Bilateral gamma distributions and processes in financial mathematics," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 261-283, February.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:2:p:261-283
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    References listed on IDEAS

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    1. Ernst Eberlein & Sebastian Raible, 1999. "Term Structure Models Driven by General Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 9(1), pages 31-53, January.
    2. Ernst Eberlein & Jean Jacod, 1997. "On the range of options prices (*)," Finance and Stochastics, Springer, vol. 1(2), pages 131-140.
    3. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305, World Scientific Publishing Co. Pte. Ltd..
    4. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
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    Cited by:

    1. Küchler, Uwe & Tappe, Stefan, 2013. "Tempered stable distributions and processes," Stochastic Processes and their Applications, Elsevier, vol. 123(12), pages 4256-4293.
    2. Lucio Fiorin & Wim Schoutens, 2020. "Conic quantization: stochastic volatility and market implied liquidity," Quantitative Finance, Taylor & Francis Journals, vol. 20(4), pages 531-542, April.
    3. Pérez-Abreu, Victor & Stelzer, Robert, 2014. "Infinitely divisible multivariate and matrix Gamma distributions," Journal of Multivariate Analysis, Elsevier, vol. 130(C), pages 155-175.
    4. Yoshihiro Shirai, 2023. "Acceptable Bilateral Gamma Parameters," Papers 2301.05333, arXiv.org.
    5. Comte, F. & Genon-Catalot, V., 2009. "Nonparametric estimation for pure jump Lévy processes based on high frequency data," Stochastic Processes and their Applications, Elsevier, vol. 119(12), pages 4088-4123, December.
    6. Matthias R. Fengler & Alexander Melnikov, 2018. "GARCH option pricing models with Meixner innovations," Review of Derivatives Research, Springer, vol. 21(3), pages 277-305, October.
    7. Küchler, Uwe & Tappe, Stefan, 2008. "On the shapes of bilateral Gamma densities," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2478-2484, October.
    8. Uehara, Yuma, 2019. "Statistical inference for misspecified ergodic Lévy driven stochastic differential equation models," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 4051-4081.
    9. Victor Korolev & Alexander Zeifman, 2023. "Quasi-Exponentiated Normal Distributions: Mixture Representations and Asymmetrization," Mathematics, MDPI, vol. 11(17), pages 1-14, September.
    10. Schneider, Paul, 2015. "Generalized risk premia," Journal of Financial Economics, Elsevier, vol. 116(3), pages 487-504.
    11. Yoshihiro Shirai, 2022. "Extreme Measures in Continuous Time Conic Finace," Papers 2210.13671, arXiv.org, revised Oct 2023.
    12. Dilip B. Madan & Wim Schoutens, 2020. "Self‐similarity in long‐horizon returns," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1368-1391, October.
    13. Kais Hamza & Fima C. Klebaner & Zinoviy Landsman & Ying-Oon Tan, 2014. "Option Pricing for Symmetric L\'evy Returns with Applications," Papers 1402.1554, arXiv.org.
    14. Filipović, Damir & Mayerhofer, Eberhard & Schneider, Paul, 2013. "Density approximations for multivariate affine jump-diffusion processes," Journal of Econometrics, Elsevier, vol. 176(2), pages 93-111.
    15. Giorgia Callegaro & Lucio Fiorin & Martino Grasselli, 2019. "Quantization meets Fourier: a new technology for pricing options," Annals of Operations Research, Springer, vol. 282(1), pages 59-86, November.
    16. Maha A. Omair & Yusra A. Tashkandy & Sameh Askar & Abdulhamid A. Alzaid, 2022. "Family of Distributions Derived from Whittaker Function," Mathematics, MDPI, vol. 10(7), pages 1-23, March.
    17. Xie, Haibin & Wang, Shouyang & Lu, Zudi, 2018. "The behavioral implications of the bilateral gamma process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 500(C), pages 259-264.
    18. 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.
    19. Küchler, Uwe & Tappe, Stefan, 2014. "Exponential stock models driven by tempered stable processes," Journal of Econometrics, Elsevier, vol. 181(1), pages 53-63.
    20. Dilip B. Madan & King Wang, 2023. "The valuation of corporations: a derivative pricing perspective," Annals of Finance, Springer, vol. 19(1), pages 1-21, March.
    21. Tomy, Lishamol & Jose, K.K., 2009. "Generalized normal-Laplace AR process," Statistics & Probability Letters, Elsevier, vol. 79(14), pages 1615-1620, July.
    22. Dilip B. Madan & Wim Schoutens & King Wang, 2020. "Bilateral multiple gamma returns: Their risks and rewards," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 7(01), pages 1-27, March.
    23. Ernst Eberlein & Antonis Papapantoleon & Albert Shiryaev, 2008. "On the duality principle in option pricing: semimartingale setting," Finance and Stochastics, Springer, vol. 12(2), pages 265-292, April.
    24. Dilip B. Madan & King Wang, 2022. "Two sided efficient frontiers at multiple time horizons," Annals of Finance, Springer, vol. 18(3), pages 327-353, September.

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