Exact Terminal Condition Neural Network for American Option Pricing Based on the Black-Scholes-Merton Equations

2025-10-30

AI Summary

Solving the Black–Scholes–Merton (BSM) partial differential equation for American option pricing is challenging due to inequality constraints arising from early exercise, strict satisfaction of terminal conditions, and the nonsmoothness and singularity of the payoff function at maturity. Method: This paper proposes a physics-informed neural network (PINN) framework featuring a terminal-condition-embedded architecture that explicitly enforces the maturity boundary; a customized loss function to handle the free-boundary constraint; and input normalization to enhance stability in high-dimensional settings. Contribution/Results: Experiments demonstrate that the proposed method achieves significantly higher accuracy than conventional finite-difference methods and state-of-the-art machine learning approaches across multiple scenarios. It exhibits strong robustness, scalability, and computational efficiency—establishing a novel, reliable paradigm for high-dimensional American option pricing.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

This paper proposes the Exact Terminal Condition Neural Network (ETCNN), a deep learning framework for accurately pricing American options by solving the Black-Scholes-Merton (BSM) equations. The ETCNN incorporates carefully designed functions that ensure the numerical solution not only exactly satisfies the terminal condition of the BSM equations but also matches the non-smooth and singular behavior of the option price near expiration. This method effectively addresses the challenges posed by the inequality constraints in the BSM equations and can be easily extended to high-dimensional scenarios. Additionally, input normalization is employed to maintain the homogeneity. Multiple experiments are conducted to demonstrate that the proposed method achieves high accuracy and exhibits robustness across various situations, outperforming both traditional numerical methods and other machine learning approaches.

Problem

• Accurately pricing American options using deep learning
• Solving Black-Scholes-Merton equations with inequality constraints
• Handling non-smooth option price behavior near expiration

Innovation

• Neural network with exact terminal condition
• Handles non-smooth behavior near expiration
• Extends to high-dimensional scenarios easily