Error Analysis of Deep PDE Solvers for Option Pricing
This work addresses the need for quantitative error analysis of deep PDE solvers in finance, providing insights for practitioners, but it is incremental as it evaluates existing methods without introducing new ones.
The paper tackled the problem of understanding the empirical accuracy of deep learning-based PDE solvers for option pricing, finding that it assessed the performance of Deep Galerkin Method and Time Deep Gradient Flow method in Black-Scholes and Heston models, with results including empirical convergence rates and training time dependencies on parameters like sampling stages and network architecture.
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.