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High order approximations of the log-Heston process semigroup

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  • Aur'elien Alfonsi
  • Edoardo Lombardo

Abstract

We present weak approximations schemes of any order for the Heston model that are obtained by using the method developed by Alfonsi and Bally (2021). This method consists in combining approximation schemes calculated on different random grids to increase the order of convergence. We apply this method with either the Ninomiya-Victoir scheme (2008) or a second-order scheme that samples exactly the volatility component, and we show rigorously that we can achieve then any order of convergence. We give numerical illustrations on financial examples that validate the theoretical order of convergence, and present also promising numerical results for the multifactor/rough Heston model.

Suggested Citation

  • Aur'elien Alfonsi & Edoardo Lombardo, 2024. "High order approximations of the log-Heston process semigroup," Papers 2407.17151, arXiv.org.
    • Handle: RePEc:arx:papers:2407.17151
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    References listed on IDEAS

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    1. Aur'elien Alfonsi, 2023. "Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation," Papers 2302.07758, arXiv.org, revised Oct 2024.
    2. Syoiti Ninomiya & Nicolas Victoir, 2008. "Weak Approximation of Stochastic Differential Equations and Application to Derivative Pricing," Applied Mathematical Finance, Taylor & Francis Journals, vol. 15(2), pages 107-121.
    3. Eduardo Abi Jaber & Omar El Euch, 2019. "Multi-factor approximation of rough volatility models," Post-Print hal-01697117, HAL.
    4. Roger Lord & Remmert Koekkoek & Dick Van Dijk, 2010. "A comparison of biased simulation schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 10(2), pages 177-194.
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    6. 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.
    7. Aurélien Alfonsi, 2015. "Affine Diffusions and Related Processes: Simulation, Theory and Applications," Post-Print hal-03127212, HAL.
    8. 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.
    9. Christian Bayer & Simon Breneis, 2023. "Weak Markovian Approximations of Rough Heston," Papers 2309.07023, arXiv.org.
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    12. Aur'elien Alfonsi & Edoardo Lombardo, 2022. "High order approximations of the Cox-Ingersoll-Ross process semigroup using random grids," Papers 2209.13334, arXiv.org, revised Apr 2023.
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