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The Weak Convergence Rate of Two Semi-Exact Discretization Schemes for the Heston Model

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  • Annalena Mickel

    (DFG Research Training Group 1953, University of Mannheim, B6, 26, D-68131 Mannheim, Germany
    Mathematical Institute, University of Mannheim, B6, 26, D-68131 Mannheim, Germany)

  • Andreas Neuenkirch

    (Mathematical Institute, University of Mannheim, B6, 26, D-68131 Mannheim, Germany)

Abstract

Inspired by the article Weak Convergence Rate of a Time-Discrete Scheme for the Heston Stochastic Volatility Model, Chao Zheng, SIAM Journal on Numerical Analysis 2017, 55:3, 1243–1263 , we studied the weak error of discretization schemes for the Heston model, which are based on exact simulation of the underlying volatility process. Both for an Euler- and a trapezoidal-type scheme for the log-asset price, we established weak order one for smooth payoffs without any assumptions on the Feller index of the volatility process. In our analysis, we also observed the usual trade off between the smoothness assumption on the payoff and the restriction on the Feller index. Moreover, we provided error expansions, which could be used to construct second order schemes via extrapolation. In this paper, we illustrate our theoretical findings by several numerical examples.

Suggested Citation

  • Annalena Mickel & Andreas Neuenkirch, 2021. "The Weak Convergence Rate of Two Semi-Exact Discretization Schemes for the Heston Model," Risks, MDPI, vol. 9(1), pages 1-38, January.
    • Handle: RePEc:gam:jrisks:v:9:y:2021:i:1:p:23-:d:478888
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    References listed on IDEAS

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    1. Simon J. A. Malham & Anke Wiese, 2013. "Chi-Square Simulation Of The Cir Process And The Heston Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(03), pages 1-38.
    2. Mark Broadie & Özgür Kaya, 2006. "Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes," Operations Research, INFORMS, vol. 54(2), pages 217-231, April.
    3. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    4. Paul M. N. Feehan & Camelia Pop, 2011. "A Schauder approach to degenerate-parabolic partial differential equations with unbounded coefficients," Papers 1112.4824, arXiv.org, revised Aug 2013.
    5. Paul Glasserman & Kyoung-Kuk Kim, 2011. "Gamma expansion of the Heston stochastic volatility model," Finance and Stochastics, Springer, vol. 15(2), pages 267-296, June.
    6. Alos, Elisa & Ewald, Christian-Oliver, 2007. "Malliavin differentiability of the Heston volatility and applications to option pricing," MPRA Paper 3237, University Library of Munich, Germany.
    7. Christian Kahl & Peter Jackel, 2006. "Fast strong approximation Monte Carlo schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 6(6), pages 513-536.
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    Cited by:

    1. Chao Zheng & Jiangtao Pan, 2023. "Unbiased estimators for the Heston model with stochastic interest rates," Papers 2301.12072, arXiv.org, revised Aug 2023.

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