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Semimartingale and continuous-time Markov chain approximation for rough stochastic local volatility models

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  • Jingtang Ma
  • Wensheng Yang
  • Zhenyu Cui

Abstract

Rough volatility models have recently been empirically shown to provide a good fit to historical volatility time series and implied volatility smiles of SPX options. They are continuous-time stochastic volatility models, whose volatility process is driven by a fractional Brownian motion with Hurst parameter less than half. Due to the challenge that it is neither a semimartingale nor a Markov process, there is no unified method that not only applies to all rough volatility models, but also is computationally efficient. This paper proposes a semimartingale and continuous-time Markov chain (CTMC) approximation approach for the general class of rough stochastic local volatility (RSLV) models. In particular, we introduce the perturbed stochastic local volatility (PSLV) model as the semimartingale approximation for the RSLV model and establish its existence , uniqueness and Markovian representation. We propose a fast CTMC algorithm and prove its weak convergence. Numerical experiments demonstrate the accuracy and high efficiency of the method in pricing European, barrier and American options. Comparing with existing literature, a significant reduction in the CPU time to arrive at the same level of accuracy is observed.

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  • Jingtang Ma & Wensheng Yang & Zhenyu Cui, 2021. "Semimartingale and continuous-time Markov chain approximation for rough stochastic local volatility models," Papers 2110.08320, arXiv.org, revised Oct 2021.
    • Handle: RePEc:arx:papers:2110.08320
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    References listed on IDEAS

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    1. Abi Jaber, Eduardo & El Euch, Omar, 2019. "Markovian structure of the Volterra Heston model," Statistics & Probability Letters, Elsevier, vol. 149(C), pages 63-72.
    2. Christian Bayer & Chiheb Ben Hammouda & Raul Tempone, 2018. "Hierarchical adaptive sparse grids and quasi Monte Carlo for option pricing under the rough Bergomi model," Papers 1812.08533, arXiv.org, revised Jan 2020.
    3. Anthonie W. Van Der Stoep & Lech A. Grzelak & Cornelis W. Oosterlee, 2014. "The Heston Stochastic-Local Volatility Model: Efficient Monte Carlo Simulation," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(07), pages 1-30.
    4. Aditi Dandapani & Paul Jusselin & Mathieu Rosenbaum, 2021. "From quadratic Hawkes processes to super-Heston rough volatility models with Zumbach effect," Quantitative Finance, Taylor & Francis Journals, vol. 21(8), pages 1235-1247, August.
    5. Aur'elien Alfonsi & Ahmed Kebaier, 2021. "Approximation of Stochastic Volterra Equations with kernels of completely monotone type," Papers 2102.13505, arXiv.org, revised Mar 2022.
    6. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," The Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    7. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2018. "Volatility is rough," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 933-949, June.
    8. Eduardo Abi Jaber & Omar El Euch, 2019. "Multi-factor approximation of rough volatility models," Post-Print hal-01697117, HAL.
    9. Ryan McCrickerd & Mikko S. Pakkanen, 2018. "Turbocharging Monte Carlo pricing for the rough Bergomi model," Quantitative Finance, Taylor & Francis Journals, vol. 18(11), pages 1877-1886, November.
    10. Martin Forde & Benjamin Smith & Lauri Viitasaari, 2021. "Rough volatility, CGMY jumps with a finite history and the Rough Heston model – small-time asymptotics in the regime," Quantitative Finance, Taylor & Francis Journals, vol. 21(4), pages 541-563, April.
    11. Christian Bayer & Chiheb Ben Hammouda & Raúl Tempone, 2020. "Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model," Quantitative Finance, Taylor & Francis Journals, vol. 20(9), pages 1457-1473, September.
    12. Eduardo Abi Jaber & Omar El Euch, 2019. "Markovian structure of the Volterra Heston model," Post-Print hal-01716696, HAL.
    13. Ludovic Goudenège & Andrea Molent & Antonino Zanette, 2020. "Machine learning for pricing American options in high-dimensional Markovian and non-Markovian models," Quantitative Finance, Taylor & Francis Journals, vol. 20(4), pages 573-591, April.
    14. 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.
    15. Christian Bayer & Peter K. Friz & Paul Gassiat & Jorg Martin & Benjamin Stemper, 2020. "A regularity structure for rough volatility," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 782-832, July.
    16. Giulia Livieri & Saad Mouti & Andrea Pallavicini & Mathieu Rosenbaum, 2018. "Rough volatility: Evidence from option prices," IISE Transactions, Taylor & Francis Journals, vol. 50(9), pages 767-776, September.
    17. Omar El Euch & Mathieu Rosenbaum, 2019. "The characteristic function of rough Heston models," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 3-38, January.
    18. Masaaki Fukasawa & Tetsuya Takabatake & Rebecca Westphal, 2019. "Is Volatility Rough ?," Papers 1905.04852, arXiv.org, revised May 2019.
    19. Ning Cai & Yingda Song & Steven Kou, 2015. "A General Framework for Pricing Asian Options Under Markov Processes," Operations Research, INFORMS, vol. 63(3), pages 540-554, June.
    20. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
    21. Giorgia Callegaro & Martino Grasselli & Gilles Paèes, 2021. "Fast Hybrid Schemes for Fractional Riccati Equations (Rough Is Not So Tough)," Mathematics of Operations Research, INFORMS, vol. 46(1), pages 221-254, February.
    22. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    23. Philippe Carmona & Laure Coutin & G. Montseny, 2000. "Approximation of Some Gaussian Processes," Statistical Inference for Stochastic Processes, Springer, vol. 3(1), pages 161-171, January.
    24. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    25. Ryan McCrickerd & Mikko S. Pakkanen, 2017. "Turbocharging Monte Carlo pricing for the rough Bergomi model," Papers 1708.02563, arXiv.org, revised Mar 2018.
    26. Omar Euch & Masaaki Fukasawa & Mathieu Rosenbaum, 2018. "The microstructural foundations of leverage effect and rough volatility," Finance and Stochastics, Springer, vol. 22(2), pages 241-280, April.
    27. Aditi Dandapani & Paul Jusselin & Mathieu Rosenbaum, 2019. "From quadratic Hawkes processes to super-Heston rough volatility models with Zumbach effect," Papers 1907.06151, arXiv.org, revised Jan 2021.
    28. Martino Grasselli, 2017. "The 4/2 Stochastic Volatility Model: A Unified Approach For The Heston And The 3/2 Model," Mathematical Finance, Wiley Blackwell, vol. 27(4), pages 1013-1034, October.
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