Error Analysis of Deep PDE Solvers for Option Pricing

Paper: arXiv 2505.05121

Authors: Jasper Rou

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.

Complexity vs Empirical Score

• Math Complexity: 7.0/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced mathematical concepts such as PDE discretization, Sobolev spaces, and deep learning optimizers, but lacks code/datasets for immediate backtesting; the experimental design is systematic and data-driven, focusing on empirical error analysis for practical implementation.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Assess accuracy & utility of<br>deep PDE solvers for option pricing"] --> B{"Methodology"}
  B --> C["Comparative Experiments<br>Black-Scholes & Heston Models"]
  B --> D["Algorithms Evaluated<br>Deep Galerkin Method<br>Time Deep Gradient Flow TDGF"]
  B --> E["Input Parameters Varied<br>Sampling stages, Samples, Layers, Nodes<br>TDGF: Discretization order, Time steps"]
  C --> F["Computational Process<br>Simulation & Training<br>Convergence Rate & Time Analysis"]
  D --> F
  E --> F
  F --> G["Key Findings & Outcomes<br>Empirical Convergence Rates<br>Training Time vs. Complexity<br>Actionable Insights for Implementation"]

Cited References (21)

1. Multifactor approximation of rough volatility models (2019)
2. Deep optimal stopping (2019)
3. The pricing of options and corporate liabilities (1973)
4. Deep hedging (2019)
5. Discrete gradient flow approximations of high dimensional evolution partial differential equations via deep neural networks (2023)
6. An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learning (2024) — link
7. Solving high-dimensional partial differential equations using deep learning (2018)
8. Global Convergence of Deep Galerkin and PINNs Methods for Solving Partial Differential Equations (2023) — link

3 of 21 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.