American double Heston options skip point-by-point free-boundary solving: one training run yields boundary, value and Greeks

The American double Heston model is treated as a free-boundary problem, and after training the network returns surfaces of the early exercise boundary, the value function and the Greeks rather than a single price. Earlier treatments of this model relied on asymptotic expansion, lattice methods, Fourier expansions or simulation, each producing an approximation to a price or a boundary. Under constant interest rate, value functions are compared side by side with the asymptotic expansion of Zhang and Feng and with least-squares Monte Carlo across three strikes and two maturity and variance settings.

The unknown early exercise boundary is fixed first, and a logarithmic transformation turns the free-boundary problem into a fixed-boundary one that the network fits in normalized form. Three sub-networks take time, the space and variance variables, and variance interaction terms; their outputs are combined multiplicatively through softmax-normalized non-negative weights, a multiplicative ensemble. Training uses sinh-stretched grid sampling, 14×14×14×14 under constant interest rate and 10×10×10×10×10 for the four-factor stochastic-rate case, and results degrade markedly without stretching.

The stochastic-rate version extends the model to four factors, two variances plus a CIR interest-rate process, and the network gains one input dimension. This version is compared with asymptotic expansion, a Shannon wavelet method and two least-squares Monte Carlo implementations. With a sparse correlation matrix, value functions at three strikes and three maturities stay close to those methods; with a dense correlation matrix, the gap from simulation grows.

The scope is numerical pricing of the American double Heston model under constant and stochastic interest rates, with parameters and grid sizes listed in the tables. After training, the method returns surfaces of the early exercise boundary, the value function and the Greeks, and the authors note that normalizing strike and value function allows prediction at unit strike, avoiding retraining per strike. The authors propose future work on full parametrization of variables and parameters, including correlation factors.

The authors report that predictions under the dense correlation matrix, though reasonable, are less consistent with least-squares Monte Carlo simulation than the sparse-matrix case, and suggest the dense correlation structure may strongly affect solution accuracy. They also note that large far-boundary values widen the interval and may require more training data, and that prediction points should lie inside the boundary points to avoid extrapolation.