Error Analysis of Option Pricing via Deep PDE Solvers: Empirical Study

Paper: arXiv 2311.07231

Abstract

Option pricing, a fundamental problem in finance, often requires solving non-linear partial differential equations (PDEs). When dealing with multi-asset options, such as rainbow options, these PDEs become high-dimensional, leading to challenges posed by the curse of dimensionality. While deep learning-based PDE solvers have recently emerged as scalable solutions to this high-dimensional problem, their empirical and quantitative accuracy remains not well-understood, hindering their real-world applicability. In this study, we aimed to offer actionable insights into the utility of Deep PDE solvers for practical option pricing implementation. Through comparative experiments, we assessed the empirical performance of these solvers in high-dimensional contexts. Our investigation identified three primary sources of errors in Deep PDE solvers: (i) errors inherent in the specifications of the target option and underlying assets, (ii) errors originating from the asset model simulation methods, and (iii) errors stemming from the neural network training. Through ablation studies, we evaluated the individual impact of each error source. Our results indicate that the Deep BSDE method (DBSDE) is superior in performance and exhibits robustness against variations in option specifications. In contrast, some other methods are overly sensitive to option specifications, such as time to expiration. We also find that the performance of these methods improves inversely proportional to the square root of batch size and the number of time steps. This observation can aid in estimating computational resources for achieving desired accuracies with Deep PDE solvers.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 8.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced high-dimensional PDE/BSDE theory and deep learning optimization, indicating high mathematical complexity, while its rigorous comparative experiments, ablation studies, and quantitative error scaling analysis provide strong empirical evidence suitable for implementation.

Research Flowchart

  flowchart TD
  Start(["Research Goal: Assess Deep PDE Solvers for<br>Option Pricing in High Dimensions"]) --> Inputs
  
  Inputs["Data/Inputs:<br>Multi-Asset Options e.g., Rainbow Options<br>Asset Model Simulations"] --> Methodology
  
  Methodology["Methodology:<br>Comparative Experiments &<br>Ablation Studies"] --> Processes
  
  Processes["Computational Processes:<br>1. Deep PDE Solvers e.g., DBSDE<br>2. Neural Network Training<br>3. Model Evaluation"] --> Findings
  
  Findings["Key Findings/Outcomes:<br>1. Identified 3 Error Sources<br>2. DBSDE is most robust & accurate<br>3. Accuracy improves ∝ 1/√Batch Size & Time Steps"]

Cited References (45)

1. An overview on deep learning-based approximation methods for partial differential equations (2020) — link
2. Neural networks-based algorithms for stochastic control and PDE s in finance (2021) — link
3. Multilevel Picard iterations for solving smooth semilinear parabolic heat equations (2021)
4. On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations (2019)
5. Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations (2020)
6. Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations (2017)
7. Solving high-dimensional partial differential equations using deep learning (2018)
8. Solving parametric PDE problems with artificial neural networks (2021)

0 of 45 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.