A physics-constrained network prices a two-asset American basket put, but its best run still overshoots Monte Carlo by about 14%
Synopsis
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(Nd) 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.
Interpretation
Pricing a multi-asset American option no longer requires writing down the early-exercise boundary in advance. Single-asset cases had been handled with finite differences, binomial trees, front-fixing or radial-basis-function quasi-interpolation, but extending such knot methods to d assets by tensor product needs O(Nd) knots and quickly becomes intractable. The single-asset machinery does not extend efficiently to multivariate domains, and the banded-matrix structure that makes it fast is not readily kept in a standard multivariate quasi-interpolation framework. In the two-asset American basket put implementation the network input grows only linearly with the number of assets, and collocation points are sampled from a low-discrepancy sequence rather than fixed on a grid.
The option value is represented by a tanh-activated feed-forward network trained so its output satisfies the pricing equation, the terminal payoff, boundary conditions and the early-exercise constraint together. Automatic differentiation supplies the second derivatives the multi-asset Black–Scholes operator needs, so the free boundary is not parameterized explicitly. The two-asset case uses a 3→64→64→64→1 network with a hand-written forward pass in MATLAB R2019b, checked against Longstaff–Schwartz least-squares Monte Carlo as an independent reference.
Getting the complementarity condition right mattered more than tuning architecture, sampling or the training schedule. Early configurations applied a two-sided squared penalty to the PDE residual over the whole domain and consistently overpriced; Run 3 gave 12.5841 against the Longstaff–Schwartz reference of 6.9005. With a one-sided residual penalty, a smooth Fischer–Burmeister complementarity function, analytic derivatives, log-moneyness coordinates and kink-line-targeted sampling, an 8000-epoch run gave 7.8669, an overestimate of 0.97, about 14%; the summary labels this configuration Run 6 while the run-by-run record lists it as Run 8.
Perspective
The demonstrated scope is a single two-asset American basket put case, implemented with a hand-written network in MATLAB R2019b and checked against Longstaff–Schwartz least-squares Monte Carlo as an independent reference.
The closest configuration still sits about 14% above the Longstaff–Schwartz reference of 6.9005, and that result comes from one of eight training setups; the summary labels it Run 6 while the run-by-run record lists it as Run 8, so the two labels disagree. Higher asset counts and other option structures are not reported in the shown scope.