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Approximating Lévy Processes With A View To Option Pricing

Author

Listed:
  • JOHN CROSBY

    (Department of Economics, University of Glasgow, UK)

  • NOLWENN LE SAUX

    (Department of Mathematics, Imperial College London, UK)

  • ALEKSANDAR MIJATOVIĆ

    (Department of Mathematics, Imperial College London, UK)

Abstract

We examine how to approximate a Lévy process by a hyperexponential jump-diffusion (HEJD) process, composed of Brownian motion and of an arbitrary number of sums of compound Poisson processes with double exponentially distributed jumps. This approximation will facilitate the pricing of exotic options since HEJD processes have a degree of tractability that other Lévy processes do not have. The idea behind this approximation has been applied to option pricing by Asmussen et al. (2007) and Jeannin and Pistorius (2008). In this paper we introduce a more systematic methodology for constructing this approximation which allow us to compute the intensity rates, the mean jump sizes and the volatility of the approximating HEJD process (almost) analytically. Our methodology is very easy to implement. We compute vanilla option prices and barrier option prices using the approximating HEJD process and we compare our results to those obtained from other methodologies in the literature. We demonstrate that our methodology gives very accurate option prices and that these prices are more accurate than those obtained from existing methodologies for approximating Lévy processes by HEJD processes.

Suggested Citation

  • John Crosby & Nolwenn Le Saux & Aleksandar Mijatović, 2010. "Approximating Lévy Processes With A View To Option Pricing," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 13(01), pages 63-91.
    • Handle: RePEc:wsi:ijtafx:v:13:y:2010:i:01:n:s0219024910005681
    DOI: 10.1142/S0219024910005681
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    References listed on IDEAS

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    1. Svetlana I Boyarchenko & Sergei Z Levendorskii, 2002. "Non-Gaussian Merton-Black-Scholes Theory," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 4955.
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    Cited by:

    1. Walter Farkas & Ludovic Mathys & Nikola Vasiljevi'c, 2020. "Intra-Horizon Expected Shortfall and Risk Structure in Models with Jumps," Papers 2002.04675, arXiv.org, revised Jan 2021.
    2. Markus Leippold & Nikola Vasiljević, 2020. "Option-Implied Intrahorizon Value at Risk," Management Science, INFORMS, vol. 66(1), pages 397-414, January.
    3. Daniel Hackmann, 2017. "Analytic techniques for option pricing under a hyperexponential L\'{e}vy model," Papers 1705.05934, arXiv.org.
    4. Walter Farkas & Ludovic Mathys & Nikola Vasiljević, 2021. "Intra‐Horizon expected shortfall and risk structure in models with jumps," Mathematical Finance, Wiley Blackwell, vol. 31(2), pages 772-823, April.
    5. Oleg Kudryavtsev, 2024. "A simplified Wiener–Hopf factorization method for pricing double barrier options under Lévy processes," Computational Management Science, Springer, vol. 21(1), pages 1-30, June.

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