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On the role of skewness and kurtosis in tempered stable (CGMY) Lévy models in finance

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  • Søren Asmussen

    (Aarhus University)

Abstract

We study the structure and properties of an infinite-activity CGMY Lévy process X $X$ with given skewness S $S$ and kurtosis K $K$ of X 1 $X_{1}$ , without a Brownian component, but allowing a drift component. The jump part of such a process is specified by the Lévy density which is C e − M x / x 1 + Y $C\mathrm {e}^{-Mx}/x^{1+Y}$ for x > 0 $x>0$ and C e − G | x | / | x | 1 + Y $C\mathrm {e}^{-G|x|}/|x|^{1+Y}$ for x

Suggested Citation

  • Søren Asmussen, 2022. "On the role of skewness and kurtosis in tempered stable (CGMY) Lévy models in finance," Finance and Stochastics, Springer, vol. 26(3), pages 383-416, July.
    • Handle: RePEc:spr:finsto:v:26:y:2022:i:3:d:10.1007_s00780-022-00482-x
    DOI: 10.1007/s00780-022-00482-x
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    1. Jérémy Poirot & Peter Tankov, 2006. "Monte Carlo Option Pricing for Tempered Stable (CGMY) Processes," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 13(4), pages 327-344, December.
    2. Young Shin Kim, 2021. "Sample Path Generation of the Stochastic Volatility CGMY Process and Its Application to Path-Dependent Option Pricing," JRFM, MDPI, vol. 14(2), pages 1-18, February.
    3. Küchler, Uwe & Tappe, Stefan, 2013. "Tempered stable distributions and processes," Stochastic Processes and their Applications, Elsevier, vol. 123(12), pages 4256-4293.
    4. Marc Jeannin & Martijn Pistorius, 2010. "A transform approach to compute prices and Greeks of barrier options driven by a class of Levy processes," Quantitative Finance, Taylor & Francis Journals, vol. 10(6), pages 629-644.
    5. Patrik Karlsson, 2018. "Finite element based Monte Carlo simulation of options on Lévy driven assets," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 5(01), pages 1-23, March.
    6. Young Shin Kim, 2021. "Sample path generation of the stochastic volatility CGMY process and its application to path-dependent option pricing," Papers 2101.11001, arXiv.org.
    7. Chengwei Zhang & Zhiyuan Zhang, 2017. "Sequential Sampling for CGMY Processes via Decomposition of their Time Changes," Papers 1708.00189, arXiv.org, revised Aug 2018.
    8. Fred Espen Benth & Jūratė Šaltytė-Benth, 2004. "The Normal Inverse Gaussian Distribution And Spot Price Modelling In Energy Markets," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 7(02), pages 177-192.
    9. Eric Ghysels & Fangfang Wang, 2014. "Moment-Implied Densities: Properties and Applications," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 32(1), pages 88-111, January.
    10. Peter Carr & Hélyette Geman & Dilip B. Madan & Marc Yor, 2003. "Stochastic Volatility for Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 13(3), pages 345-382, July.
    11. Bianchi, Michele Leonardo & Rachev, Svetlozar T. & Kim, Young Shin & Fabozzi, Frank J., 2011. "Tempered infinitely divisible distributions and processes," Working Paper Series in Economics 26, Karlsruhe Institute of Technology (KIT), Department of Economics and Management.
    12. Fomichov, Vladimir & González Cázares, Jorge & Ivanovs, Jevgenijs, 2021. "Implementable coupling of Lévy process and Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 407-431.
    13. R. A. Doney & R. A. Maller, 2002. "Stability and Attraction to Normality for Lévy Processes at Zero and at Infinity," Journal of Theoretical Probability, Springer, vol. 15(3), pages 751-792, July.
    14. Laura Ballotta & Ioannis Kyriakou, 2014. "Monte Carlo Simulation of the CGMY Process and Option Pricing," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 34(12), pages 1095-1121, December.
    15. 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.
    16. Rohatgi, Vijay K. & Székely, Gábor J., 1989. "Sharp inequalities between skewness and kurtosis," Statistics & Probability Letters, Elsevier, vol. 8(4), pages 297-299, September.
    17. Søren Asmussen & Mogens Bladt, 2022. "Gram–Charlier methods, regime-switching and stochastic volatility in exponential Lévy models," Quantitative Finance, Taylor & Francis Journals, vol. 22(4), pages 675-689, April.
    18. Rimas Norvaiša & Donna Mary Salopek, 2002. "Estimating the p-Variation Index of a Sample Function: An Application to Financial Data Set," Methodology and Computing in Applied Probability, Springer, vol. 4(1), pages 27-53, March.
    19. Chengwei Zhang & Zhiyuan Zhang, 2018. "Sequential sampling for CGMY processes via decomposition of their time changes," Naval Research Logistics (NRL), John Wiley & Sons, vol. 65(6-7), pages 522-534, September.
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    1. Gero Junike, 2023. "On the number of terms in the COS method for European option pricing," Papers 2303.16012, arXiv.org, revised Mar 2024.

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    More about this item

    Keywords

    Cumulant; Functional limit theorem; Log-return distribution; Exponentially tilted stable distribution; Moment method; Wasserstein distance;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions

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