Finance Informed Neural Networks with Residual Gain Attention Based Radial Basis Functions

In the Black–Scholes framework, pricing a European call differs fundamentally from pricing an American put. A European call can only be exercised at maturity and admits the classic closed-form Black–Scholes solution, whereas the American put’s early-exercise feature turns its governing PDE into a free-boundary problem, requiring simultaneous determination of both the option value and the optimal exercise boundary and thus greatly increasing numerical complexity. The RGA-RBF FINN model addresses these challenges by embedding the PDE directly into its architecture via RBF-based feature extraction, residual connections, and an attention mechanism for dynamic focus. By incorporating the PDE residual into the loss function, the network enforces theoretical constraints while capturing nonlinear dynamics and free-boundary behavior. Empirical results show that RGA-RBF FINN not only recovers the closed-form valuation of European calls and solves the free-boundary problem for American puts, but also delivers superior computational efficiency and pricing accuracy in scenarios with irregular boundaries or sparse data.