This paper applies Physics-Informed Neural Networks (PINNs) to price European call and American put options under the Black-Scholes framework. The PINN model incorporates the PDE residual, boundary conditions, and initial conditions into a unified loss function, enabling data-driven learning while respecting physical laws. The authors test their approach on both simulated data (with known analytical or numerical solutions) and real market data from Optionmetrics and Yahoo Finance, focusing on NDX European calls and SPXC American puts.

Key methodological innovations include a gated network architecture that combines a simple linear path with a deeper nonlinear path to capture the ramp-like structure of option payoffs, as well as the use of tanh activation and residual connections to improve gradient flow. On simulated data, the model achieves near-perfect correlation with benchmarks (MSE 0.052 for European call, 0.23 for American put), though it exhibits some deviations near boundaries and underestimation of high prices for American puts. On market data, the PINN model outperforms analytical benchmarks, reducing RMSE by approximately 30% for both option types, with correlations of 0.63 (NDX call) and 0.73 (SPXC put) to market prices.

The paper also discusses convergence and stability issues, noting oscillatory behavior during training and sensitivity to the PDE penalty weight. Overall, the study demonstrates the potential of PINNs for option pricing, particularly for American options where analytical solutions are infeasible, and highlights areas for further improvement such as better handling of boundary conditions and free-boundary problems.