American Option Pricing Based on Coupled PINN: A New Curriculum Learning Strategy Under the Heston Model

This article introduces a project published by Rohan-217 on GitHub (link: https://github.com/Rohan-217/American_Options_Pricing_using_Coupled_PINNs, related paper: https://arxiv.org/abs/2605.06688, release date: June 2, 2026). Its core is to use coupled Physics-Informed Neural Networks (Coupled PINN) combined with curriculum learning and adaptive resampling techniques to solve the pricing problem of American put options under the Heston stochastic volatility model, achieving joint prediction of option prices and optimal exercise boundaries.

American options allow early exercise, so two coupled problems need to be solved simultaneously: calculating the option price and determining the time-varying optimal exercise boundary. The Heston model describes stochastic volatility more realistically than the market, but there is no closed-form solution for American options under this model, and numerical methods are relied upon for solving, which increases the difficulty of pricing.

The project uses a coupled PINN architecture, which differs from traditional separate processing methods. It can learn the price function and exercise boundary simultaneously, capturing the inherent relationship between them. PINN embeds the Heston stochastic partial differential equation as a physical constraint into the loss function, enabling physically reasonable predictions even when data is sparse.

To improve model stability, two techniques are introduced: 1. Curriculum learning: Start training from simple regions far from the exercise boundary, and gradually introduce complex samples near the boundary to avoid local optima; 2. Adaptive resampling: Dynamically adjust the distribution of training points, increasing sampling density in areas with large errors (such as the exercise boundary) to effectively capture regions with sharp gradient changes.

The project provides two usage methods: 1. Jupyter Notebook: End-to-end complete implementation, suitable for quick understanding and experiments; 2. Modular code (heson_pinn folder): Separates data generation, network definition, training loop, and visualization for easy expansion. The coupled network design allows two sub-networks to output price and boundary estimates respectively, maintaining physical consistency through shared layers and a joint loss function.

Experiments verify the effectiveness of the method. Compared with traditional numerical methods (finite difference, Monte Carlo), it has three major advantages: 1. Fast inference speed: After training, pricing only requires forward propagation with millisecond-level response; 2. Continuously differentiable: Facilitates calculation of sensitivity indicators such as Greek letters; 3. Strong generalization ability: Can perform interpolation predictions for parameter configurations outside the training set. It can accurately estimate prices and exercise boundaries.

This scheme shifts the computational burden from 'per pricing' to 'one-time training', improving the feasibility of real-time pricing and demonstrating the potential of PINN in financial engineering. It can be extended to pricing complex derivatives such as multi-asset options and path-dependent options. The combined strategy of curriculum learning and adaptive resampling also provides reference for physics-informed learning of other free boundary problems.