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

F Bourgey, S de Marco, Emmanuel Gobet (emmanuel.gobet@polytechnique.edu) and Alexandre Zhou
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F Bourgey: CMAP - Centre de Mathématiques Appliquées de l'Ecole polytechnique - Inria - Institut National de Recherche en Informatique et en Automatique - 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 - Inria - Institut National de Recherche en Informatique et en Automatique - 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 - Inria - Institut National de Recherche en Informatique et en Automatique - 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 nationale des ponts et chaussées

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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.

Date: 2020-04-15
Note: View the original document on HAL open archive server: https://hal.science/hal-02430430v1
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Published in Monte Carlo Methods and Applications, 2020, 26 (2), ⟨10.1515/mcma-2020-2062⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-02430430

DOI: 10.1515/mcma-2020-2062

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