Error Analysis of Deep PDE Solvers for Option Pricing

TL;DR

The paper addresses the practical accuracy of deep PDE solvers for option pricing by comparing the Time Deep Gradient Flow (TDGF) and Deep Galerkin Method (DGM) on Black–Scholes and Heston PDEs. It systematically varies five training factors—sampling stages, samples, network depth, width, and TDGF time steps—and reports $L^2$-error and training time, finding near-linear error decline with sampling stages and clear benefits from second-order time stepping for TDGF. The work provides actionable guidance: use more sampling stages until diminishing returns set in, prefer larger widths (40–50) for TDGF, keep layers around 2–3, and adopt second-order time discretization for TDGF; DGM shows less consistent sensitivity to samples and width. Overall, the results quantify trade-offs and offer practical recommendations to deploy deep PDE solvers in option pricing settings with higher confidence and efficiency.

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.

Figures (19)

• Figure 1: Exact price of European call option with $r=0.05$, $\sigma=0.25$ and $T=1.0$ compared to the price of computed by the TDGF.
• Figure 2: $L^2$-error of the TDGF for a call option in the Black--Scholes model against number of sampling stages varying from 16 to 500.
• Figure 3: $L^2$-error of the TDGF for a call option in the Heston model against number of sampling stages varying from 16 to 500.
• Figure 4: $L^2$-error of the DGM for a call option in the Black--Scholes model against number of sampling stages varying from 2048 to 100,000.
• Figure 5: $L^2$-error of the DGM for a call option in the Heston model against number of sampling stages varying from 2048 to 100,000.