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Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation

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  • Michael B. Giles
  • Kristian Debrabant
  • Andreas Ro{ss}ler

Abstract

The multilevel Monte Carlo path simulation method introduced by Giles ({\it Operations Research}, 56(3):607-617, 2008) exploits strong convergence properties to improve the computational complexity by combining simulations with different levels of resolution. In this paper we analyse its efficiency when using the Milstein discretisation; this has an improved order of strong convergence compared to the standard Euler-Maruyama method, and it is proved that this leads to an improved order of convergence of the variance of the multilevel estimator. Numerical results are also given for basket options to illustrate the relevance of the analysis.

Suggested Citation

  • Michael B. Giles & Kristian Debrabant & Andreas Ro{ss}ler, 2013. "Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation," Papers 1302.4676, arXiv.org, revised Jun 2019.
  • Handle: RePEc:arx:papers:1302.4676
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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. Rainer Avikainen, 2009. "On irregular functionals of SDEs and the Euler scheme," Finance and Stochastics, Springer, vol. 13(3), pages 381-401, September.
    3. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
    4. Dereich, Steffen & Heidenreich, Felix, 2011. "A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1565-1587, July.
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    Cited by:

    1. Devang Sinha & Siddhartha P. Chakrabarty, 2022. "Multilevel Richardson-Romberg and Importance Sampling in Derivative Pricing," Papers 2209.00821, arXiv.org.
    2. Mouna Ben Derouich & Ahmed Kebaier, 2022. "The interpolated drift implicit Euler scheme Multilevel Monte Carlo method for pricing Barrier options and applications to the CIR and CEV models," Papers 2210.00779, arXiv.org.
    3. Michael B. Giles & Abdul-Lateef Haji-Ali, 2019. "Sub-sampling and other considerations for efficient risk estimation in large portfolios," Papers 1912.05484, arXiv.org, revised Apr 2022.
    4. Nabil Kahale, 2018. "General multilevel Monte Carlo methods for pricing discretely monitored Asian options," Papers 1805.09427, arXiv.org, revised Sep 2018.
    5. Michael B. Giles & Abdul-Lateef Haji-Ali, 2022. "Multilevel Path Branching for Digital Options," Papers 2209.03017, arXiv.org.
    6. Kahalé, Nabil, 2020. "General multilevel Monte Carlo methods for pricing discretely monitored Asian options," European Journal of Operational Research, Elsevier, vol. 287(2), pages 739-748.

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