High-dimensional option pricing and hedging present significant challenges in quantitative finance, where traditional PDE-based methods struggle with the curse of dimensionality. The BSDE framework offers a computationally efficient alternative to PDE-based methods, and recently proposed deep BSDE solvers, generally utilizing conventional Multi-Layer Perceptrons (MLPs), build upon this framework to provide a scalable alternative to numerical BSDE solvers. In this research, we show that although such MLP-based deep BSDEs demonstrate promising results in option pricing, there remains room for improvement regarding hedging performance. To address this issue, we introduce KANHedge, a novel BSDE-based hedger that leverages Kolmogorov-Arnold Networks (KANs) within the BSDE framework. Unlike conventional MLP approaches that use fixed activation functions, KANs employ learnable B-spline activation functions that provide enhanced function approximation capabilities for continuous derivatives. We comprehensively evaluate KANHedge on both European and American basket options across multiple dimensions and market conditions. Our experimental results demonstrate that while KANHedge and MLP achieve comparable pricing accuracy, KANHedge provides improved hedging performance. Specifically, KANHedge achieves considerable reductions in hedging cost metrics, demonstrating enhanced risk control capabilities.

翻译：高维期权定价与对冲在量化金融领域面临重大挑战，传统基于偏微分方程的方法深受维度诅咒的困扰。倒向随机微分方程框架为基于偏微分方程的方法提供了一种计算高效的替代方案，而近期提出的深度倒向随机微分方程求解器通常采用传统的多层感知机，基于该框架为数值倒向随机微分方程求解器提供了可扩展的替代方案。本研究表明，尽管此类基于多层感知机的深度倒向随机微分方程在期权定价方面展现出良好效果，但其对冲性能仍有提升空间。为解决此问题，我们提出KANHedge——一种基于倒向随机微分方程框架的新型对冲模型，该模型在倒向随机微分方程框架中引入了Kolmogorov-Arnold网络。与使用固定激活函数的传统多层感知机方法不同，KAN采用可学习的B样条激活函数，从而为连续导数提供了更强的函数逼近能力。我们在多种维度与市场条件下，对欧式和美式篮子期权进行了全面评估。实验结果表明，虽然KANHedge与多层感知机在定价精度上表现相当，但KANHedge提供了更优的对冲性能。具体而言，KANHedge在对冲成本指标上实现了显著降低，展现出更强的风险控制能力。