A Monotone Limit Approach to Entropy-Regularized American Options

TL;DR

This paper introduces a probabilistic approach using entropy regularization and reflected backward stochastic differential equations to approximate American option values, providing explicit convergence rates and a new policy improvement algorithm.

Contribution

It presents a novel probabilistic framework for entropy-regularized American options using Doob-Meyer decomposition and reflected BSDEs, with proven convergence and a practical policy algorithm.

Findings

Monotone approximation of the value function with explicit convergence rates

Development of a policy improvement algorithm based on linear BSDEs

Numerical illustration on American max call option demonstrates effectiveness

Abstract

Recent advances in continuous-time optimal stopping have been driven by entropy-regularized formulations of randomized stopping problems, with most existing approaches relying on partial differential equation methods. In this paper, we propose a fully probabilistic framework based on the Doob-Meyer-Mertens decomposition of the Snell envelope and its representation through reflected backward stochastic differential equations. We introduce an entropy-regularized penalization scheme yielding a monotone approximation of the value function and establish explicit convergence rates under suitable regularity assumptions. In addition, we develop a policy improvement algorithm based on linear backward stochastic differential equations and illustrate its performance through a simple numerical experiment for an American-style max call option