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Multilevel Monte-Carlo methods and lower-upper bounds in Initial Margin computations

Author

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  • F Bourgey

    (CMAP - Centre de Mathématiques Appliquées de l'Ecole polytechnique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique)

  • S de Marco

    (CMAP - Centre de Mathématiques Appliquées de l'Ecole polytechnique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique)

  • Emmanuel Gobet

    (CMAP - Centre de Mathématiques Appliquées de l'Ecole polytechnique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique)

  • Alexandre Zhou

    (CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique - ENPC - École des Ponts ParisTech)

Abstract

The Multilevel Monte-Carlo (MLMC) method developed by Giles [Gil08] has a natural application to the evaluation of nested expectation of the form E [g(E [f (X, Y)|X])], where f, g are functions and (X, Y) a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of Initial Margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotical optimality; at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal/dual algorithms for stochastic control problems.

Suggested Citation

  • F Bourgey & S de Marco & Emmanuel Gobet & Alexandre Zhou, 2020. "Multilevel Monte-Carlo methods and lower-upper bounds in Initial Margin computations," Post-Print hal-02430430, HAL.
    • Handle: RePEc:hal:journl:hal-02430430
    DOI: 10.1515/mcma-2020-2062
    Note: View the original document on HAL open archive server: https://hal.science/hal-02430430
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    References listed on IDEAS

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    1. Rainer Avikainen, 2009. "On irregular functionals of SDEs and the Euler scheme," Finance and Stochastics, Springer, vol. 13(3), pages 381-401, September.
    2. Mark Broadie & Yiping Du & Ciamac C. Moallemi, 2015. "Risk Estimation via Regression," Operations Research, INFORMS, vol. 63(5), pages 1077-1097, October.
    3. Michael B. Gordy & Sandeep Juneja, 2010. "Nested Simulation in Portfolio Risk Measurement," Management Science, INFORMS, vol. 56(10), pages 1833-1848, October.
    4. Ivan Guo & Gregoire Loeper, 2018. "Pricing Bounds for Volatility Derivatives via Duality and Least Squares Monte Carlo," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 598-617, November.
    5. Michael B. Giles & Abdul-Lateef Haji-Ali, 2018. "Multilevel nested simulation for efficient risk estimation," Papers 1802.05016, arXiv.org, revised Feb 2019.
    6. K. Bujok & B. M. Hambly & C. Reisinger, 2015. "Multilevel Simulation of Functionals of Bernoulli Random Variables with Application to Basket Credit Derivatives," Methodology and Computing in Applied Probability, Springer, vol. 17(3), pages 579-604, September.
    7. Mark Broadie & Paul Glasserman, 1996. "Estimating Security Price Derivatives Using Simulation," Management Science, INFORMS, vol. 42(2), pages 269-285, February.
    8. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
    9. Martin B. Haugh & Leonid Kogan, 2004. "Pricing American Options: A Duality Approach," Operations Research, INFORMS, vol. 52(2), pages 258-270, April.
    10. repec:hal:wpaper:hal-01686952 is not listed on IDEAS
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    Cited by:

    1. Alfonsi, Aurélien & Cherchali, Adel & Infante Acevedo, Jose Arturo, 2021. "Multilevel Monte-Carlo for computing the SCR with the standard formula and other stress tests," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 234-260.
    2. Aur'elien Alfonsi & Adel Cherchali & Jose Arturo Infante Acevedo, 2020. "Multilevel Monte-Carlo for computing the SCR with the standard formula and other stress tests," Papers 2010.12651, arXiv.org, revised Apr 2021.

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