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Solving high-dimensional optimal stopping problems using deep learning

Part of: Stochastic processes Mathematical finance Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  27 April 2021

SEBASTIAN BECKER ,
PATRICK CHERIDITO ,
ARNULF JENTZEN  and
TIMO WELTI Open the ORCID record for TIMO WELTI [Opens in a new window]
SEBASTIAN BECKER
Affiliation:
RiskLab, Department of Mathematics, ETH Zürich, 8092Zürich, Switzerland emails:[email protected]; [email protected]
PATRICK CHERIDITO
Affiliation:
RiskLab, Department of Mathematics, ETH Zürich, 8092Zürich, Switzerland emails:[email protected]; [email protected]
ARNULF JENTZEN
Affiliation:
Seminar for Applied Mathematics, Department of Mathematics, ETH Zürich, 8092Zürich, Switzerland and Faculty of Mathematics and Computer Science, University of Münster, 48149Münster, Germany email: [email protected]
TIMO WELTI
Affiliation:
Seminar for Applied Mathematics, Department of Mathematics, ETH Zürich, 8092Zürich, Switzerland and D ONE Solutions AG, 8003Zürich, Switzerland email: [email protected]
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Abstract

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Nowadays many financial derivatives, such as American or Bermudan options, are of early exercise type. Often the pricing of early exercise options gives rise to high-dimensional optimal stopping problems, since the dimension corresponds to the number of underlying assets. High-dimensional optimal stopping problems are, however, notoriously difficult to solve due to the well-known curse of dimensionality. In this work, we propose an algorithm for solving such problems, which is based on deep learning and computes, in the context of early exercise option pricing, both approximations of an optimal exercise strategy and the price of the considered option. The proposed algorithm can also be applied to optimal stopping problems that arise in other areas where the underlying stochastic process can be efficiently simulated. We present numerical results for a large number of example problems, which include the pricing of many high-dimensional American and Bermudan options, such as Bermudan max-call options in up to 5000 dimensions. Most of the obtained results are compared to reference values computed by exploiting the specific problem design or, where available, to reference values from the literature. These numerical results suggest that the proposed algorithm is highly effective in the case of many underlyings, in terms of both accuracy and speed.

Keywords

American optionBermudan optionfinancial derivativederivative pricingoption pricingoptimal stoppingcurse of dimensionalitydeep learning

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 65C05: Monte Carlo methods 91G60: Numerical methods (including Monte Carlo methods)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Special Issue 3: Connections between Deep learning and Partial Differential Equations , June 2021 , pp. 470 - 514
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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