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.

翻译：期权定价是金融领域的基础问题，通常需要求解非线性偏微分方程（PDE）。当处理多资产期权（如彩虹期权）时，这些PDE会呈现高维特性，从而引发维数灾难的挑战。尽管基于深度学习的PDE求解器近期已成为应对此类高维问题的可扩展方案，但其经验性与定量精度仍不明确，制约了其实际应用。本研究旨在为深度PDE求解器在期权定价实践中的效用提供可操作见解。通过对比实验，我们评估了这些求解器在高维场景下的经验性能。研究发现深度PDE求解器的三类主要误差来源：（i）目标期权与基础资产规格的固有误差，（ii）资产模型模拟方法带来的误差，（iii）神经网络训练产生的误差。通过消融实验，我们量化了各误差源的独立影响。结果表明，深度BSDE方法（DBSDE）性能优越，且对期权规格变化具有稳健性；而其他方法对期权规格（如到期时间）过度敏感。此外，我们发现这些方法的性能随批大小平方根倒数与时间步数倒数呈正相关改进，该发现有助于估算实现深度PDE求解器预期精度所需的计算资源。