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

ArXiv ID: 2311.07231 “View on arXiv”

Authors: Unknown

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.

Keywords: option pricing, partial differential equations (PDE), deep learning, curse of dimensionality, Deep BSDE, Derivatives (Multi-Asset Options)

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 8.0/10
  • Quadrant: Holy Grail
  • Why: 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.
  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"]