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Importance sampling for option pricing with feedforward neural networks

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  • Aleksandar Arandjelovi'c
  • Thorsten Rheinlander
  • Pavel V. Shevchenko

Abstract

We study the problem of reducing the variance of Monte Carlo estimators through performing suitable changes of the sampling measure which are induced by feedforward neural networks. To this end, building on the concept of vector stochastic integration, we characterize the Cameron-Martin spaces of a large class of Gaussian measures which are induced by vector-valued continuous local martingales with deterministic covariation. We prove that feedforward neural networks enjoy, up to an isometry, the universal approximation property in these topological spaces. We then prove that sampling measures which are generated by feedforward neural networks can approximate the optimal sampling measure arbitrarily well. We conclude with a comprehensive numerical study pricing path-dependent European options for asset price models that incorporate factors such as changing business activity, knock-out barriers, dynamic correlations, and high-dimensional baskets.

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  • Aleksandar Arandjelovi'c & Thorsten Rheinlander & Pavel V. Shevchenko, 2021. "Importance sampling for option pricing with feedforward neural networks," Papers 2112.14247, arXiv.org, revised Jun 2023.
    • Handle: RePEc:arx:papers:2112.14247
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    1. Baldi, P. & Ben Arous, G. & Kerkyacharian, G., 1992. "Large deviations and the Strassen theorem in Hölder norm," Stochastic Processes and their Applications, Elsevier, vol. 42(1), pages 171-180, August.
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