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Solving high-dimensional optimal stopping problems using deep learning

by S. Becker and P. Cheridito and A. Jentzen and T. Welti

(Report number 2019-60)

Abstract
Nowadays many financial derivatives which are traded on stock and futures exchanges, 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 underlyings in the associated hedging portfolio. 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 for 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, for example, 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 option, Bermudan option, financial derivative, derivative pricing, option pricing, optimal stopping, curse of dimensionality, deep learning

@Techreport{BCJW19_864,
  author = {S. Becker and P. Cheridito and A. Jentzen and T. Welti},
  title = {Solving high-dimensional optimal stopping problems using deep learning},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-60},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-60.pdf },
  year = {2019}
}

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