Pricing and Hedging American-Style Options with Deep Learning

Sebastian Becker,Patrick Cheridito,Arnulf Jentzen

doi:10.3390/jrfm13070158

Sebastian Becker, Patrick Cheridito + Show 1 more

Open Access

https://doi.org/10.3390/jrfm13070158

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Abstract

In this paper we introduce a deep learning method for pricing and hedging American-style options. It first computes a candidate optimal stopping policy. From there it derives a lower bound for the price. Then it calculates an upper bound, a point estimate and confidence intervals. Finally, it constructs an approximate dynamic hedging strategy. We test the approach on different specifications of a Bermudan max-call option. In all cases it produces highly accurate prices and dynamic hedging strategies with small replication errors.

Highlights

Exercise options are notoriously difficult to value
In Sirignano and Spiliopoulos (2018) optimal stopping problems in continuous time have been solved by approximating the solutions of the corresponding free boundary PDEs with deep neural networks
In order to do that we introduce the functions vθn (x) := g(n, x) ∨ cθn (x), Cθn (x) := 0 ∨ cθn (x), x ∈ Rd, and hedge the difference vθn (YnM) − Cθn−1 (Y(n−1)M) on each of the time intervals [tn−1, tn], n = 1, . . . , N, separately. vθn describes the approximate value of the option at time tn if it has not been exercised before, and the definition of Cθn takes into account that the true continuation values are non-negative due to the non-negativity of the payoff function g

Summary

Introduction

For up to three underlying risk factors, tree based and classical PDE approximation methods usually yield good numerical results; see, e.g., Forsyth and Vetzal (2002); Hull (2003); Reisinger and Witte (2012) and the references therein. Haugh and Kogan (2004) as well as Kohler et al (2010) have already used shallow neural networks to estimate continuation values. In Sirignano and Spiliopoulos (2018) optimal stopping problems in continuous time have been solved by approximating the solutions of the corresponding free boundary PDEs with deep neural networks. The main focus of these papers is to derive optimal stopping rules and accurate price estimates

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Conclusion