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Computations of Greeks in a market with jumps via the Malliavin calculus

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  • Youssef El-Khatib
  • Nicolas Privault

Abstract

Using the Malliavin calculus on Poisson space we compute Greeks in a market driven by a discontinuous process with Poisson jump times and random jump sizes, following a method initiated on the Wiener space in [5]. European options do not satisfy the regularity conditions required in our approach, however we show that Asian options can be considered due to a smoothing effect of the integral over time. Numerical simulations are presented for the Delta and Gamma of Asian options, and confirm the efficiency of this approach over classical finite difference Monte-Carlo approximations of derivatives. Copyright Springer-Verlag Berlin/Heidelberg 2004

Suggested Citation

  • Youssef El-Khatib & Nicolas Privault, 2004. "Computations of Greeks in a market with jumps via the Malliavin calculus," Finance and Stochastics, Springer, vol. 8(2), pages 161-179, May.
    • Handle: RePEc:spr:finsto:v:8:y:2004:i:2:p:161-179
    DOI: 10.1007/s00780-003-0111-6
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    Citations

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    Cited by:

    1. Hyungbin Park, 2018. "Sensitivity analysis of long-term cash flows," Finance and Stochastics, Springer, vol. 22(4), pages 773-825, October.
    2. Barbara Forster & Eva Luetkebohmert & Josef Teichmann, 2005. "Absolutely continuous laws of Jump-Diffusions in finite and infinite dimensions with applications to mathematical Finance," Papers math/0509016, arXiv.org, revised Oct 2008.
    3. Reiichiro Kawai, 2012. "Likelihood ratio gradient estimation for Meixner distribution and Lévy processes," Computational Statistics, Springer, vol. 27(4), pages 739-755, December.
    4. El-Khatib, Youssef & Goutte, Stephane & Makumbe, Zororo S. & Vives, Josep, 2023. "A hybrid stochastic volatility model in a Lévy market," International Review of Economics & Finance, Elsevier, vol. 85(C), pages 220-235.
    5. Kawai, Reiichiro & Takeuchi, Atsushi, 2010. "Sensitivity analysis for averaged asset price dynamics with gamma processes," Statistics & Probability Letters, Elsevier, vol. 80(1), pages 42-49, January.
    6. Privault, Nicolas & Wei, Xiao, 2004. "A Malliavin calculus approach to sensitivity analysis in insurance," Insurance: Mathematics and Economics, Elsevier, vol. 35(3), pages 679-690, December.
    7. Atsushi Takeuchi, 2010. "Bismut–Elworthy–Li-Type Formulae for Stochastic Differential Equations with Jumps," Journal of Theoretical Probability, Springer, vol. 23(2), pages 576-604, June.
    8. Anastasis Kratsios, 2019. "Partial Uncertainty and Applications to Risk-Averse Valuation," Papers 1909.13610, arXiv.org, revised Oct 2019.
    9. Masafumi Hayashi, 2010. "Coefficients of Asymptotic Expansions of SDE with Jumps," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 17(4), pages 373-389, December.
    10. El-Khatib, Youssef & Abdulnasser, Hatemi-J, 2011. "On the calculation of price sensitivities with jump-diffusion structure," MPRA Paper 30596, University Library of Munich, Germany.
    11. Anselm Hudde & Ludger Rüschendorf, 2023. "European and Asian Greeks for Exponential Lévy Processes," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-24, March.
    12. Yeliz Yolcu-Okur & Tilman Sayer & Bilgi Yilmaz & B. Alper Inkaya, 2018. "Computation of the Delta of European options under stochastic volatility models," Computational Management Science, Springer, vol. 15(2), pages 213-237, June.
    13. Davis, Mark H.A. & Johansson, Martin P., 2006. "Malliavin Monte Carlo Greeks for jump diffusions," Stochastic Processes and their Applications, Elsevier, vol. 116(1), pages 101-129, January.
    14. Ayub Ahmadi & Mahdieh Tahmasebi, 2024. "Pricing and delta computation in jump-diffusion models with stochastic intensity by Malliavin calculus," Papers 2405.00473, arXiv.org.
    15. Bilgi Yilmaz, 2018. "Computation of option greeks under hybrid stochastic volatility models via Malliavin calculus," Papers 1806.06061, arXiv.org.

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