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

TL;DR

AM-PIRN targets nonlinear option pricing PDEs with sharp gradients by adaptively redistributing collocation points according to PDE residuals while preserving a fixed total point budget, and replaces standard FC networks with a ResNet backbone to stabilize training. It solves equations of the form $U_t + \mathcal{N}[U] = 0$ by moving movable points according to $p(x;\Theta) \propto \frac{|r^k(x;\Theta)|}{\mathbb{E}[|r^k(x;\Theta)|]}$ and retaining immovable points. Across generalized Black-Scholes, Barles–Soner, CEV, and Heston models, AM-PIRN achieves lower PDE residuals and competitive option price accuracy versus PINN, RAM-PINN, and WAM-PINN, with particularly strong performance in multi-dimensional settings. This work advances robust, adaptive numerical pricing for complex derivatives and supports potential extensions to stochastic volatility, path-dependent payoffs, and uncertainty quantification.

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.

Figures (12)

• Figure 1: The workflow of AM-PIRN: Adaptive movement sampling Physics-informed residual network. $\mathcal{X}_{p,i}$ is the fixed set of collocation points that remains constant during the model training, generated using a uniform sampling strategy. $\mathcal{X}_{p,m}$ is the set of movable collocation points that will dynamically change during the training process.
• Figure 2: The exact solution and numerical solutions of PINN, RAM-PINN, WAM-PINN and AM-PIRN of the general Black-Scholes equation (\ref{['eq:19']}).
• Figure 3: The PDE residuals (first row), absolute errors of estimated solutions (middle row), and distribution of mobile collocation points (last row) for the general Black-Scholes equation (\ref{['eq:19']}) computed by PINN, RAM-PINN, WAM-PINN and AM-PIRN after $10$ iteration rounds, respectively.
• Figure 4: The iterative convergence curves of the $L_2$ errors of the estimated solutions for the general Black-Scholes equation by PINN, RAM-PINN, WAM-PINN and AM-PIRN.
• Figure 5: The numerical solutions of Barles’ and Soner’s model given by PINN, RAM-PINN, WAM-PINN and AM-PIRN, respectively.