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An explicit Euler scheme with strong rate of convergence for financial SDEs with non-Lipschitz coefficients

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  • Jean-Francois Chassagneux
  • Antoine Jacquier
  • Ivo Mihaylov

Abstract

We consider the approximation of stochastic differential equations (SDEs) with non-Lipschitz drift or diffusion coefficients. We present a modified explicit Euler-Maruyama discretisation scheme that allows us to prove strong convergence, with a rate. Under some regularity and integrability conditions, we obtain the optimal strong error rate. We apply this scheme to SDEs widely used in the mathematical finance literature, including the Cox-Ingersoll-Ross~(CIR), the 3/2 and the Ait-Sahalia models, as well as a family of mean-reverting processes with locally smooth coefficients. We numerically illustrate the strong convergence of the scheme and demonstrate its efficiency in a multilevel Monte Carlo setting.

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  • Jean-Francois Chassagneux & Antoine Jacquier & Ivo Mihaylov, 2014. "An explicit Euler scheme with strong rate of convergence for financial SDEs with non-Lipschitz coefficients," Papers 1405.3561, arXiv.org, revised Apr 2016.
    • Handle: RePEc:arx:papers:1405.3561
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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. 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.
    3. Ait-Sahalia, Yacine, 1996. "Testing Continuous-Time Models of the Spot Interest Rate," The Review of Financial Studies, Society for Financial Studies, vol. 9(2), pages 385-426.
    4. 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.
    5. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
    6. Andreas Neuenkirch & Lukasz Szpruch, 2012. "First order strong approximations of scalar SDEs with values in a domain," Papers 1209.0390, arXiv.org.
    7. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
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    Cited by:

    1. Mario Hefter & Arnulf Jentzen, 2019. "On arbitrarily slow convergence rates for strong numerical approximations of Cox–Ingersoll–Ross processes and squared Bessel processes," Finance and Stochastics, Springer, vol. 23(1), pages 139-172, January.
    2. Cui, Zhenyu & Kirkby, J. Lars & Nguyen, Duy, 2021. "Efficient simulation of generalized SABR and stochastic local volatility models based on Markov chain approximations," European Journal of Operational Research, Elsevier, vol. 290(3), pages 1046-1062.
    3. Ngo, Hoang Long & Luong, Duc Trong, 2019. "Tamed Euler–Maruyama approximation for stochastic differential equations with locally Hölder continuous diffusion coefficients," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 133-140.
    4. Yifan Bai & Xing Huang, 2023. "Log-Harnack Inequality and Exponential Ergodicity for Distribution Dependent Chan–Karolyi–Longstaff–Sanders and Vasicek Models," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1902-1921, September.
    5. Blanka Horvath & Oleg Reichmann, 2018. "Dirichlet Forms and Finite Element Methods for the SABR Model," Papers 1801.02719, arXiv.org.
    6. Gao, Xiangyu & Wang, Jianqiao & Wang, Yanxia & Yang, Hongfu, 2022. "The truncated Euler–Maruyama method for CIR model driven by fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 189(C).
    7. C'onall Kelly & Gabriel J. Lord, 2021. "An adaptive splitting method for the Cox-Ingersoll-Ross process," Papers 2112.09465, arXiv.org, revised Feb 2023.
    8. 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.

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