American option pricing with model constrained Gaussian process regressions

This article introduces a novel method based on Gaussian process regression for pricing American options. The variational partial differential equation (PDE) governing option prices is converted into a non-linear penalized Feynman-Kac equation (PFK). We propose an iterative algorithm to manage the non-linearity of the PFK operator. We sample state variables in the PDE’ s inner domain and on the terminal boundary. At each step, we fit a constrained regression function approximating the option price. This function matches the option payoffs on the boundary sample while satisfying the PFK PDE on the inner sample. The non-linear term in this PDE is frozen and valued with the price estimate from the previous iteration. We adopt a Bayesian framework in which payoffs and the value of the FK PDE in the boundary and inner samples are noised. Assuming the regression function is a Gaussian process, we find a closed-form approximation of option prices. In the numerical illustration, we evaluate American put options in the Heston, Black-Scholes models and in the two-factor Hull-White model.