Physics-Informed Neural Networks with Adaptive Sampling for Option Pricing

Abstract

Today, machine learning has an accelerated impact in quantitative finance. Current models require large amounts of data, which can be expensive. A notable area of research, physics-informed neural networks (PINNs), has proven to be effective in approximating problems that are described by partial differential equations (PDEs). During training, the PDE is embedded in the loss function and evaluated at the residual points. This allows these types of neural networks to solve problems where data is scarce or noisy. Recent studies have shown that the method for sampling residual points has a great influence on training efficiency. Residual-based adaptive distribution (RAD) sampling is the adaptive sampling method used throughout this paper. This research applies PINNs with RAD sampling to solve the Black-Scholes PDE. Here, the Black-Scholes model is used to determine the price of options in a financial market. The fundamental goal of this paper is to study the difference in training performance between non-adaptive and RAD sampling. The types of options that are being considered in this study, are the European call options and the American put options. The results shown suggest that both types of options benefit from using RAD sampling compared to non-adaptive sampling. With a loss decrease of 39.33\%, American put options improve more using RAD sampling than European call options. Although European call options still show a decrease in loss of 7.57\%.