European option pricing with model constrained Gaussian process regressions

Abstract

We propose a method for pricing European options based on Gaussian processes. We convert the problem of solving the Feynman-Kac (FK) partial differential equation (PDE) into a model-constrained regression. We form two training sets by sampling state variables from the PDEs inner domain and terminal boundary. The regression function is then estimated to fit the option payoffs on the boundary sample while satisfying the FK PDE on the inner sample. 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 closedform approximation for the option prices. We demonstrate the performance of the procedure on call options in the Heston model and basket call options in a Black-Scholes market.

Keywords: Gaussian process regression, Option pricing, Feynman-Kac equation, partial differential equation, Heston model, machine learning