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Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation

Author

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  • Ahmed Kebaier

    (Université Paris 13)

  • Jérôme Lelong

    (University Grenoble Alpes)

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 handle 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," Methodology and Computing in Applied Probability, Springer, vol. 20(2), pages 611-641, June.
    • Handle: RePEc:spr:metcap:v:20:y:2018:i:2:d:10.1007_s11009-017-9579-y
    DOI: 10.1007/s11009-017-9579-y
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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. 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.
    3. 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.
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    Citations

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    Cited by:

    1. Devang Sinha & Siddhartha P. Chakrabarty, 2022. "Multilevel Monte Carlo and its Applications in Financial Engineering," Papers 2209.14549, arXiv.org.
    2. Jérôme Lelong & Zineb El Filali Ech-Chafiq & Adil Reghai, 2020. "Automatic Control Variates for Option Pricing using Neural Networks," Working Papers hal-02891798, HAL.
    3. 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.
    4. Jérôme Lelong & Zineb El Filali Ech-Chafiq & Adil Reghai, 2021. "Automatic Control Variates for Option Pricing using Neural Networks," Post-Print hal-02891798, HAL.
    5. Devang Sinha & Siddhartha P. Chakrabarty, 2022. "Multilevel Richardson-Romberg and Importance Sampling in Derivative Pricing," Papers 2209.00821, 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.
    7. Devang Sinha & Siddhartha P. Chakrabarty, 2024. "Multilevel Monte Carlo in Sample Average Approximation: Convergence, Complexity and Application," Papers 2407.18504, arXiv.org.

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