Neural networks-based algorithms for option pricing

Abstract

In this research, we consider neural network-algorithms for option pricing. We use the Black-Scholes model and the lifted Heston model. We derive the option pricing partial differential equation (PDE), which we solve with a neural network, and the conditional characteristic function of the stock price which leads to the option price with the COS method. We consider two neural network-algorithms: the Deep Galerkin Method (DGM) and the Time Deep Nitsche Method (TDNM). We extend the TDNM to be able to solve the option pricing PDE by splitting the PDE operator in a symmetric part and an asymmetric part. The splitting method is more stable than transforming the Black-Scholes option pricing PDE to the symmetric heat equation. The DGM can predict options prices perfectly in the Black-Scholes model even when $r$ and $\sigma$ are added as variables to the neural network. In the lifted Heston model, the DGM can predict option prices perfectly for small dimensions, but has larger errors for larger dimensions. The TDNM is as good as DGM for small times to maturity and volatilities, but has larger errors for large times to maturity and volatilities.