Error Analysis of Deep PDE Solvers for Option Pricing

TL;DR

This paper evaluates the empirical accuracy and convergence of deep learning-based PDE solvers, specifically the Deep Galerkin Method and TDGF, for option pricing models like Black-Scholes and Heston.

Contribution

It provides a detailed empirical analysis of deep PDE solvers' performance, including convergence rates and training times, under various hyperparameter settings.

Findings

Deep PDE solvers show varying convergence rates depending on hyperparameters.

Training time increases with the number of samples and layers.

Empirical accuracy is influenced by discretization scheme and number of time steps.

Abstract

Option pricing often requires solving partial differential equations (PDEs). Although deep learning-based PDE solvers have recently emerged as quick solutions to this problem, their empirical and quantitative accuracy remain not well understood, hindering their real-world applicability. In this research, our aim is to offer actionable insights into the utility of deep PDE solvers for practical option pricing implementation. Through comparative experiments in both the Black--Scholes and the Heston model, we assess the empirical performance of two neural network algorithms to solve PDEs: the Deep Galerkin Method and the Time Deep Gradient Flow method (TDGF). We determine their empirical convergence rates and training time as functions of (i) the number of sampling stages, (ii) the number of samples, (iii) the number of layers, and (iv) the number of nodes per layer. For the TDGF, we also consider the order of the discretization scheme and the number of time steps.