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

期权定价是金融学中的一个基本问题，通常需要解决非线性偏微分方程（PDE）的问题。当处理多资产期权（如彩虹期权）时，这些PDE变成高维的，从而产生了维数灾难的挑战。虽然基于深度学习的PDE求解器最近出现作为这个高维问题的可扩展解决方案，但它们的经验和数量精度仍不为人们所了解，从而阻碍了它们在实际应用中的适用性。在这项研究中，我们旨在为深度PDE求解器在实际期权定价实现中的效用提供可行的见解。通过比较实验，我们评估了这些求解器在高维情境下的经验表现。我们的调查确定了深度PDE求解器中的三个主要误差来源：（i）目标期权和基础资产规格中固有的误差，（ii）源于资产模型模拟方法的误差，以及（iii）源于神经网络训练的误差。通过消融研究，我们评估了每个误差来源的个体影响。我们的结果表明，深度BSDE方法（DBSDE）在性能上优于其他方法，并且展现出对期权规格变化的鲁棒性。相反，其他一些方法对期权规格（如到期时间）过于敏感。我们还发现，这些方法的性能与批量大小和时间步数的平方根成反比。这一观察结果可以帮助估计使用深度PDE求解器实现所需精度的计算资源。

The paper aims to assess the empirical performance of deep learning-based PDE solvers in high-dimensional contexts, specifically in option pricing, and identify the sources of errors in these solvers.

The key idea of the paper is to evaluate the performance of different deep learning-based PDE solvers in high-dimensional contexts for option pricing and identify the sources of errors in these solvers, including errors inherent in the specifications of the target option and underlying assets, errors originating from the asset model simulation methods, and errors stemming from the neural network training.

The paper conducts comparative experiments to evaluate the empirical performance of deep learning-based PDE solvers in high-dimensional contexts for option pricing and identifies three primary sources of errors in these solvers. The Deep BSDE method (DBSDE) is found to be superior in performance and exhibits robustness against variations in option specifications. The paper also finds that the performance of these methods improves inversely proportional to the square root of batch size and the number of time steps. The paper's experiment is designed using synthetic datasets, and the code is made publicly available.

Related research in this field includes 'Deep Learning for Option Pricing, Hedging and Greeks Approximation' by Bühler et al., 'Neural Networks for Option Pricing' by Guglielmi et al., and 'A Deep Learning Framework for Financial Time Series using Stacked Autoencoders and Long Short-Term Memory' by Zheng et al.