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Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation

Author

Listed:
  • Ahmed Kebaier

    (LAGA - Laboratoire Analyse, Géométrie et Applications - UP8 - Université Paris 8 Vincennes-Saint-Denis - UP13 - Université Paris 13 - Institut Galilée - CNRS - Centre National de la Recherche Scientifique)

  • Jérôme Lelong

    (DAO - Données, Apprentissage et Optimisation - LJK - Laboratoire Jean Kuntzmann - Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology - Inria - Institut National de Recherche en Informatique et en Automatique - CNRS - Centre National de la Recherche Scientifique - UGA [2016-2019] - Université Grenoble Alpes [2016-2019])

Abstract

In this work, we propose a smart idea to couple importance sampling and Multilevel Monte Carlo (MLMC). We advocate a per level approach with as many importance sampling parameters as the number of levels, which enables us to compute the different levels independently. The search for parameters is carried out using sample average approximation, which basically consists in applying deterministic optimisation techniques to a Monte Carlo approximation rather than resorting to stochastic approximation. Our innovative estimator leads to a robust and efficient procedure reducing both the discretization error (the bias) and the variance for a given computational effort. In the setting of discretized diffusions, we prove that our estimator satisfies a strong law of large numbers and a central limit theorem with optimal limiting variance, in the sense that this is the variance achieved by the best importance sampling measure (among the class of changes we consider), which is however non tractable. Finally, we illustrate the efficiency of our method on several numerical challenges coming from quantitative finance and show that it outperforms the standard MLMC estimator.

Suggested Citation

  • Ahmed Kebaier & Jérôme Lelong, 2018. "Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation," Post-Print hal-01214840, HAL.
    • Handle: RePEc:hal:journl:hal-01214840
    DOI: 10.1007/s11009-017-9579-y
    Note: View the original document on HAL open archive server: https://hal.science/hal-01214840v4
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    References listed on IDEAS

    as
    1. Michael B. Giles & Lukasz Szpruch, 2012. "Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without L\'{e}vy area simulation," Papers 1202.6283, arXiv.org, revised May 2014.
    2. Lelong, Jérôme, 2008. "Almost sure convergence of randomly truncated stochastic algorithms under verifiable conditions," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2632-2636, November.
    3. 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.
    4. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
    5. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Jérôme Lelong & Zineb El Filali Ech-Chafiq & Adil Reghai, 2021. "Automatic Control Variates for Option Pricing using Neural Networks," Post-Print hal-02891798, HAL.
    2. Devang Sinha & Siddhartha P. Chakrabarty, 2022. "Multilevel Monte Carlo and its Applications in Financial Engineering," Papers 2209.14549, arXiv.org.
    3. Devang Sinha & Siddhartha P. Chakrabarty, 2022. "Multilevel Richardson-Romberg and Importance Sampling in Derivative Pricing," Papers 2209.00821, arXiv.org.
    4. 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.
    5. Jérôme Lelong & Zineb El Filali Ech-Chafiq & Adil Reghai, 2020. "Automatic Control Variates for Option Pricing using Neural Networks," Working Papers hal-02891798, HAL.
    6. Mouna Ben Derouich & Ahmed Kebaier, 2022. "Interpolated Drift Implicit Euler MLMC Method for Barrier Option Pricing and application to CIR and CEV Models," Papers 2210.00779, arXiv.org, revised Sep 2024.

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