Deep learning and American options via free boundary framework

TL;DR

This paper introduces a deep learning approach for American options pricing that effectively captures the free boundary, utilizing a novel auxiliary function and Landau transformation to improve accuracy and efficiency.

Contribution

The paper presents a new deep learning framework incorporating a dual solution and auxiliary function to solve American options with free boundary features.

Findings

Accurately approximates early exercise boundary and its derivative.

Demonstrates efficiency and competitiveness against existing numerical methods.

Provides reliable estimates of option Greeks from the neural network output.

Abstract

We propose a deep learning method for solving the American options model with a free boundary feature. To extract the free boundary known as the early exercise boundary from our proposed method, we introduce the Landau transformation. For efficient implementation of our proposed method, we further construct a dual solution framework consisting of a novel auxiliary function and free boundary equations. The auxiliary function is formulated to include the feed forward deep neural network (DNN) output and further mimic the far boundary behaviour, smooth pasting condition, and remaining boundary conditions due to the second-order space derivative and first-order time derivative. Because the early exercise boundary and its derivative are not a priori known, the boundary values mimicked by the auxiliary function are in approximate form. Concurrently, we then establish equations that approximate the early exercise boundary and its derivative directly from the DNN output based on some linear relationships at the left boundary. Furthermore, the option Greeks are obtained from the derivatives of this auxiliary function. We test our implementation with several examples and compare them with the existing numerical methods. All indicators show that our proposed deep learning method presents an efficient and alternative way of pricing options with early exercise features.