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A Monotone Limit Approach to Entropy-Regularized American Options

Daniel Chee, Noufel Frikha, Libo Li

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

A Monotone Limit Approach to Entropy-Regularized American Options

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
Paper Structure (7 sections, 7 theorems, 82 equations, 1 table)

This paper contains 7 sections, 7 theorems, 82 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Entropy-Regularized Penalization Scheme
  4. Definition of the entropy-regularized penalization scheme
  5. Convergence to the American Option
  6. Policy Improvement Algorithm
  7. Numerical Experiments

Key Result

Proposition 3.1

For any $\lambda \in (0,1]$, there exists a unique solution $(v^\lambda, m^\lambda) \in \mathcal{S}^2 \times {\mathcal{M}}^2$ to dongbsdemod.

Theorems & Definitions (16)

  • Remark 3.1
  • Proposition 3.1
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.2
  • ...and 6 more