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Automatic Control Variates for Option Pricing using Neural Networks

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  • Jérôme Lelong

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

  • Zineb El Filali Ech-Chafiq

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

  • Adil Reghai

    (Natixis Asset Management)

Abstract

Many pricing problems boil down to the computation of a high dimensional integral, which is usually estimated using Monte Carlo. In fact, the accuracy of a Monte Carlo estimator with M simulations is given by σ √ M. Meaning that its convergence is immune to the dimension of the problem. However, this convergence can be relatively slow depending on the variance σ of the function to be integrated. To resolve such a problem, one would perform some variance reduction techniques such as importance sampling, stratification, or control variates. In this paper, we will study two approaches for improving the convergence of Monte Carlo using Neural Networks. The first approach relies on the fact that many high dimensional financial problems are of low effective dimensions[15]. We expose a method to reduce the dimension of such problems in order to keep only the necessary variables. The integration can then be done using fast numerical integration techniques such as Gaussian quadrature. The second approach consists in building an automatic control variate using neural networks. We learn the function to be integrated (which incorporates the diffusion model plus the payoff function) in order to build a network that is highly correlated to it. As the network that we use can be integrated exactly, we can use it as a control variate.

Suggested Citation

  • Jérôme Lelong & Zineb El Filali Ech-Chafiq & Adil Reghai, 2021. "Automatic Control Variates for Option Pricing using Neural Networks," Post-Print hal-02891798, HAL.
  • Handle: RePEc:hal:journl:hal-02891798
    DOI: 10.1515/mcma-2020-2081
    Note: View the original document on HAL open archive server: https://hal.univ-grenoble-alpes.fr/hal-02891798
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    References listed on IDEAS

    as
    1. Ahmed Kebaier & Jérôme Lelong, 2018. "Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation," Post-Print hal-01214840, HAL.
    2. P. Pellizzari, 2001. "Efficient Monte Carlo pricing of European options¶using mean value control variates," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 24(2), pages 107-126, November.
    3. 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.
    4. Portier, Francois & Segers, Johan, 2019. "Monte Carlo integration with a growing number of control variates," LIDAM Reprints ISBA 2019035, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    5. Sujin Kim & Shane G. Henderson, 2007. "Adaptive Control Variates for Finite-Horizon Simulation," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 508-527, August.
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