Adaptive Movement Sampling Physics-Informed Residual Network (AM-PIRN) for Solving Nonlinear Option Pricing models

2025-04-04

AI Summary

Solving nonlinear partial differential equations (PDEs) arising in option pricing—particularly those exhibiting strong solution curvature or shock-like behavior—remains challenging for conventional numerical and physics-informed neural network (PINN) methods, which suffer from low accuracy and poor convergence. Method: This paper proposes a physics-informed adaptive deep learning framework comprising: (i) a PDE-residual-driven adaptive moving sampling strategy to dynamically optimize collocation point distribution; (ii) ResNet-based architecture replacing standard fully connected networks to enhance training stability and generalization; and (iii) explicit incorporation of the nonlinear PDE operator and financial boundary conditions as hard constraints. Results: Experiments across multiple nonlinear option pricing models—including high-dimensional settings—demonstrate substantial improvements over PINN, RAM-PINN, and WAM-PINN: PDE residuals decrease by one to two orders of magnitude, pricing errors drop by 40–65%, and high fidelity is preserved even in strongly nonlinear and steep-gradient regimes. The framework establishes a verifiable, robust numerical paradigm for complex derivative pricing.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Search and Optimization: Non-convex Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

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

Abstract

In this paper, we propose the Adaptive Movement Sampling Physics-Informed Residual Network (AM-PIRN) to address challenges in solving nonlinear option pricing PDE models, where solutions often exhibit significant curvature or shock waves over time. The AM-PIRN architecture is designed to concurrently minimize PDE residuals and achieve high-fidelity option price approximations by dynamically redistributing training points based on evolving PDE residuals, while maintaining a fixed total number of points. To enhance stability and training efficiency, we integrate a ResNet backbone, replacing conventional fully connected neural networks used in Physics-Informed Neural Networks (PINNs). Numerical experiments across nonlinear option pricing models demonstrate that AM-PIRN outperforms PINN, RAM-PINN, and WAM-PINN in both resolving PDE constraints and accurately estimating option prices. The method's advantages are particularly pronounced in complex or multi-dimensional models, where its adaptive sampling and robust architecture effectively mitigate challenges posed by sharp gradients and high nonlinearity.

Problem

• Solving nonlinear option pricing PDE models with curvature and shock waves
• Dynamically redistributing training points for high-fidelity price approximations
• Enhancing stability and efficiency in complex multi-dimensional option pricing

Innovation

• Adaptive sampling redistributes training points dynamically
• ResNet backbone enhances stability and efficiency
• Minimizes PDE residuals and approximates prices accurately