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Strong convergence rates for Euler approximations to a class of stochastic path-dependent volatility models

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  • Andrei Cozma
  • Christoph Reisinger

Abstract

We consider a class of stochastic path-dependent volatility models where the stochastic volatility, whose square follows the Cox-Ingersoll-Ross model, is multiplied by a (leverage) function of the spot price, its running maximum, and time. We propose a Monte Carlo simulation scheme which combines a log-Euler scheme for the spot process with the full truncation Euler scheme or the backward Euler-Maruyama scheme for the squared stochastic volatility component. Under some mild regularity assumptions and a condition on the Feller ratio, we establish the strong convergence with order 1/2 (up to a logarithmic factor) of the approximation process up to a critical time. The model studied in this paper contains as special cases Heston-type stochastic-local volatility models, the state-of-the-art in derivative pricing, and a relatively new class of path-dependent volatility models. The present paper is the first to prove the convergence of the popular Euler schemes with a positive rate, which is moreover consistent with that for Lipschitz coefficients and hence optimal.

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  • Andrei Cozma & Christoph Reisinger, 2017. "Strong convergence rates for Euler approximations to a class of stochastic path-dependent volatility models," Papers 1706.07375, arXiv.org, revised Oct 2018.
    • Handle: RePEc:arx:papers:1706.07375
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    References listed on IDEAS

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    1. Michael Giles & Desmond Higham & Xuerong Mao, 2009. "Analysing multi-level Monte Carlo for options with non-globally Lipschitz payoff," Finance and Stochastics, Springer, vol. 13(3), pages 403-413, September.
    2. 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.
    3. Alexander Lipton & Andrey Gal & Andris Lasis, 2014. "Pricing of vanilla and first-generation exotic options in the local stochastic volatility framework: survey and new results," Quantitative Finance, Taylor & Francis Journals, vol. 14(11), pages 1899-1922, November.
    4. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
    5. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    6. 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.
    7. Andrei Cozma & Matthieu Mariapragassam & Christoph Reisinger, 2017. "Calibration of a Hybrid Local-Stochastic Volatility Stochastic Rates Model with a Control Variate Particle Method," Papers 1701.06001, arXiv.org, revised Mar 2021.
    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. Ben Hambly & Matthieu Mariapragassam & Christoph Reisinger, 2016. "A forward equation for barrier options under the Brunick & Shreve Markovian projection," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 827-838, June.
    10. Leif Andersen & Vladimir Piterbarg, 2007. "Moment explosions in stochastic volatility models," Finance and Stochastics, Springer, vol. 11(1), pages 29-50, January.
    11. Ben Hambly & Matthieu Mariapragassam & Christoph Reisinger, 2014. "A Forward Equation for Barrier Options under the Brunick&Shreve Markovian Projection," Papers 1411.3618, arXiv.org, revised Sep 2016.
    12. Andrei Cozma & Christoph Reisinger, 2017. "Strong order 1/2 convergence of full truncation Euler approximations to the Cox-Ingersoll-Ross process," Papers 1704.07321, arXiv.org, revised Oct 2018.
    13. Alfonsi, Aurélien, 2013. "Strong order one convergence of a drift implicit Euler scheme: Application to the CIR process," Statistics & Probability Letters, Elsevier, vol. 83(2), pages 602-607.
    14. Andrei Cozma & Matthieu Mariapragassam & Christoph Reisinger, 2015. "Convergence of an Euler scheme for a hybrid stochastic-local volatility model with stochastic rates in foreign exchange markets," Papers 1501.06084, arXiv.org, revised Oct 2016.
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