Deep Penalty Methods: A Class of Deep Learning Algorithms for Solving High Dimensional Optimal Stopping Problems

TL;DR

The paper introduces Deep Penalty Method (DPM), a deep learning algorithm inspired by penalty methods for high-dimensional optimal stopping problems, validated on American option pricing.

Contribution

It develops a novel deep learning algorithm for high-dimensional optimal stopping, combining penalty methods with the Deep BSDE framework, and provides theoretical error bounds.

Findings

Error of DPM bounded by loss function and penalty parameters

Discretization error converges at rate 1/2

Numerical tests confirm accuracy and efficiency in American option pricing

Abstract

We propose a deep learning algorithm for high dimensional optimal stopping problems. Our method is inspired by the penalty method for solving free boundary PDEs. Within our approach, the penalized PDE is approximated using the Deep BSDE framework proposed by \cite{weinan2017deep}, which leads us to coin the term "Deep Penalty Method (DPM)" to refer to our algorithm. We show that the error of the DPM can be bounded by the loss function and $O(\frac{1}{\lambda})+O(\lambda h) +O(\sqrt{h})$, where $h$ is the step size in time and $\lambda$ is the penalty parameter. This finding emphasizes the need for careful consideration when selecting the penalization parameter and suggests that the discretization error converges at a rate of order $\frac{1}{2}$. We validate the efficacy of the DPM through numerical tests conducted on a high-dimensional optimal stopping model in the area of American option pricing. The numerical tests confirm both the accuracy and the computational efficiency of our proposed algorithm.