# Optimal Exercise Boundary

Pricing American options on several assets is hard because the early-exercise boundary is unknown, and extending knot methods to d assets by tensor product needs O(N^d) knots. A physics-informed neural network represents the option value with a tanh feed-forward network, uses automatic differentiation for the multi-asset Black–Scholes operator, and trains on sampled collocation points instead of a grid. On a two-asset American basket put, the 8000-epoch configuration with a one-sided residual penalty and a smooth Fischer–Burmeister complementarity function gave 7.8669, an overestimate of 0.97, about 14%, against the Longstaff–Schwartz Monte Carlo reference of 6.9005, and the closest of eight training setups.

Pricing American options on several assets is hard because the early-exercise boundary is unknown, and extending knot methods to d assets by tensor product needs O(N^d) knots. A physics-informed neural network represents the option value with a tanh feed-forward network, uses automatic differentiation for the multi-asset Black–Scholes operator, and trains on sampled collocation points instead of a grid. On a two-asset American basket put, the 8000-epoch configuration with a one-sided residual penalty and a smooth Fischer–Burmeister complementarity function gave 7.8669, an overestimate of 0.97, about 14%, against the Longstaff–Schwartz Monte Carlo reference of 6.9005, and the closest of eight training setups.