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Malliavin Monte Carlo Greeks for jump diffusions

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  • Davis, Mark H.A.
  • Johansson, Martin P.

Abstract

In recent years efficient methods have been developed for calculating derivative price sensitivities using Monte Carlo simulation. Malliavin calculus has been used to transform the simulation problem in the case where the underlying follows a Markov diffusion process. In this work, recent developments in the area of Malliavin calculus for Lévy processes are applied and slightly extended. This allows for derivation of similar stochastic weights as in the continuous case for a certain class of jump-diffusion processes.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:1:p:101-129
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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. 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.
    3. Cont, Rama & Lu, Yi, 2016. "Weak approximation of martingale representations," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 857-882.
    4. Bilgi Yilmaz, 2018. "Computation of option greeks under hybrid stochastic volatility models via Malliavin calculus," Papers 1806.06061, arXiv.org.
    5. Lars Peter Hansen, 2008. "Modeling the Long Run: Valuation in Dynamic Stochastic Economies," NBER Working Papers 14243, National Bureau of Economic Research, Inc.
    6. Muroi, Yoshifumi & Suda, Shintaro, 2017. "Computation of Greeks in jump-diffusion models using discrete Malliavin calculus," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 140(C), pages 69-93.
    7. Yuri F. Saporito, 2020. "Pricing Path-Dependent Derivatives under Multiscale Stochastic Volatility Models: a Malliavin Representation," Papers 2005.04297, arXiv.org.
    8. 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.
    9. N. Hilber & N. Reich & C. Schwab & C. Winter, 2009. "Numerical methods for Lévy processes," Finance and Stochastics, Springer, vol. 13(4), pages 471-500, September.
    10. 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.
    11. Chen, Nan & Glasserman, Paul, 2007. "Malliavin Greeks without Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 117(11), pages 1689-1723, November.
    12. Delphine David & Nicolas Privault, 2009. "Numerical computation of Theta in a jump-diffusion model by integration by parts," Quantitative Finance, Taylor & Francis Journals, vol. 9(6), pages 727-735.
    13. Mohamed Ben Alaya & Ahmed Kebaier & Ngoc Khue Tran, 2020. "Local asymptotic properties for Cox‐Ingersoll‐Ross process with discrete observations," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(4), pages 1401-1464, December.
    14. 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.
    15. 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.
    16. Leitao, Álvaro & Oosterlee, Cornelis W. & Ortiz-Gracia, Luis & Bohte, Sander M., 2018. "On the data-driven COS method," Applied Mathematics and Computation, Elsevier, vol. 317(C), pages 68-84.
    17. Fard, Farzad Alavi & Siu, Tak Kuen, 2013. "Pricing participating products with Markov-modulated jump–diffusion process: An efficient numerical PIDE approach," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 712-721.
    18. Farzad Fard & Ning Rong, 2014. "Pricing and managing risks of ruin contingent life annuities under regime switching variance gamma process," Annals of Finance, Springer, vol. 10(2), pages 315-332, May.
    19. 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.
    20. 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.
    21. Solé, Josep Lluís & Utzet, Frederic & Vives, Josep, 2007. "Canonical Lévy process and Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 117(2), pages 165-187, February.

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