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Unbiased Deep Solvers for Linear Parametric PDEs

Unbiased Deep Solvers for Linear Parametric PDEs We develop several deep learning algorithms for approximating families of parametric PDE solutions. The proposed algorithms approximate solutions together with their gradients, which in the context of mathematical finance means that the derivative prices and hedging strategies are computed simultaneously. Having approximated the gradient of the solution, one can combine it with a Monte Carlo simulation to remove the bias in the deep network approximation of the PDE solution (derivative price). This is achieved by leveraging the Martingale Representation Theorem and combining the Monte Carlo simulation with the neural network. The resulting algorithm is robust with respect to the quality of the neural network approximation and consequently can be used as a black box in case only limited a-priori information about the underlying problem is available. We believe this is important as neural network-based algorithms often require fair amount of tuning to produce satisfactory results. The methods are empirically shown to work for high-dimensional problems (e.g., 100 dimensions). We provide diagnostics that shed light on appropriate network architectures. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematical Finance Taylor & Francis

Unbiased Deep Solvers for Linear Parametric PDEs

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Unbiased Deep Solvers for Linear Parametric PDEs

Abstract

We develop several deep learning algorithms for approximating families of parametric PDE solutions. The proposed algorithms approximate solutions together with their gradients, which in the context of mathematical finance means that the derivative prices and hedging strategies are computed simultaneously. Having approximated the gradient of the solution, one can combine it with a Monte Carlo simulation to remove the bias in the deep network approximation of the PDE solution (derivative...
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Publisher
Taylor & Francis
Copyright
© 2022 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
ISSN
1466-4313
eISSN
1350-486X
DOI
10.1080/1350486X.2022.2030773
Publisher site
See Article on Publisher Site

Abstract

We develop several deep learning algorithms for approximating families of parametric PDE solutions. The proposed algorithms approximate solutions together with their gradients, which in the context of mathematical finance means that the derivative prices and hedging strategies are computed simultaneously. Having approximated the gradient of the solution, one can combine it with a Monte Carlo simulation to remove the bias in the deep network approximation of the PDE solution (derivative price). This is achieved by leveraging the Martingale Representation Theorem and combining the Monte Carlo simulation with the neural network. The resulting algorithm is robust with respect to the quality of the neural network approximation and consequently can be used as a black box in case only limited a-priori information about the underlying problem is available. We believe this is important as neural network-based algorithms often require fair amount of tuning to produce satisfactory results. The methods are empirically shown to work for high-dimensional problems (e.g., 100 dimensions). We provide diagnostics that shed light on appropriate network architectures.

Journal

Applied Mathematical Finance – Taylor & Francis

Published: Jul 4, 2021

Keywords: Monte Carlo method; deep neural network; control variates; partial differential equations; 65M75; 60H30; 91G60

References

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