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Quasi-Monte Carlo-Based Conditional Malliavin Method for Continuous-Time Asian Option Greeks

Author

Listed:
  • Chao Yu

    (Tsinghua University)

  • Xiaoqun Wang

    (Tsinghua University)

Abstract

Although many methods for computing the Greeks of discrete-time Asian options are proposed, few methods to calculate the Greeks of continuous-time Asian options are known. In this paper, we develop an integration by parts formula in the multi-dimensional Malliavin calculus, and apply it to obtain the Greeks formulae for continuous-time Asian options in the multi-asset situation. We combine the Malliavin method with the quasi-Monte Carlo method to calculate the Greeks in simulation. We discuss the asymptotic convergence of simulation estimates for the continuous-time Asian option Greeks obtained by Malliavin derivatives. We propose to use the conditional quasi-Monte Carlo method to smooth Malliavin Greeks, and show that the calculation of conditional expectations analytically is viable for many types of Asian options. We prove that the new estimates for Greeks have good smoothness. For binary Asian options, Asian call options and up-and-out Asian call options, for instance, our estimates are infinitely times differentiable. We take the gradient principal component analysis method as a dimension reduction technique in simulation. Numerical experiments demonstrate the large efficiency improvement of the proposed method, especially for Asian options with discontinuous payoff functions.

Suggested Citation

  • Chao Yu & Xiaoqun Wang, 2023. "Quasi-Monte Carlo-Based Conditional Malliavin Method for Continuous-Time Asian Option Greeks," Computational Economics, Springer;Society for Computational Economics, vol. 62(1), pages 325-360, June.
    • Handle: RePEc:kap:compec:v:62:y:2023:i:1:d:10.1007_s10614-022-10257-3
    DOI: 10.1007/s10614-022-10257-3
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    References listed on IDEAS

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    1. Chaojun Zhang & Xiaoqun Wang, 2020. "Quasi-Monte Carlo-based conditional pathwise method for option Greeks," Quantitative Finance, Taylor & Francis Journals, vol. 20(1), pages 49-67, January.
    2. Xiaoqun Wang & Ken Seng Tan, 2013. "Pricing and Hedging with Discontinuous Functions: Quasi-Monte Carlo Methods and Dimension Reduction," Management Science, INFORMS, vol. 59(2), pages 376-389, July.
    3. Ye Xiao & Xiaoqun Wang, 2019. "Enhancing Quasi-Monte Carlo Simulation by Minimizing Effective Dimension for Derivative Pricing," Computational Economics, Springer;Society for Computational Economics, vol. 54(1), pages 343-366, June.
    4. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
    5. Christian Bayer & Markus Siebenmorgen & Raul Tempone, 2018. "Smoothing the payoff for efficient computation of Basket option prices," Quantitative Finance, Taylor & Francis Journals, vol. 18(3), pages 491-505, March.
    6. Mark Broadie & Paul Glasserman, 1996. "Estimating Security Price Derivatives Using Simulation," Management Science, INFORMS, vol. 42(2), pages 269-285, February.
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    Cited by:

    1. Chao Yu & Yuhan Cheng, 2023. "Malliavin Calculus and Its Application to Robust Optimal Investment for an Insider," Mathematics, MDPI, vol. 11(20), pages 1-38, October.

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