AI 中文总结
本研究重构Carr-Madan期权定价框架，采用SFFT实现次指数计算缩放，建立连接经典傅里叶定价、张量网络与量子计算的统一框架，为高维金融计算提供可扩展方案。
AI 中文摘要
当标的资产过程的特征函数可用时，基于傅里叶的方法是对欧式期权定价最广泛使用的技术之一。然而，它们对日益精细离散化的适用性受到经典傅里叶变换内存需求快速增长的限制，这成为大规模定价问题的计算瓶颈。在本研究中，我们通过使用张量网络重构Carr-Madan定价框架克服了这一限制。具体而言，我们采用超快速傅里叶变换（SFFT），这是量子傅里叶变换（QFT）的压缩张量列车表示，并将其直接应用于张量化期权定价，无需显式构建指数级大的向量或傅里叶算子。该公式还通过在量子模拟器和量子硬件上基于QFT的期权定价，实现了经典张量网络算法与其量子对应算法之间的直接比较。对欧式看涨期权的数值实验表明，所提出的SFFT方法保持了定价精度，同时大幅降低了内存需求，并实现了比传统基于FFT的定价更优的次指数计算缩放。伴随的量子模拟和硬件执行实现了经典张量网络公式与其基于QFT的量子对应算法之间的直接比较，显示两种方法均避免了传统傅里叶实现的指数缩放，并为大规模期权定价提供了互补视角。这些结果共同建立了连接经典傅里叶定价、张量网络算法和量子计算方法的统一框架，证明了张量化傅里叶方法如何为高维金融计算提供可扩展的替代方案。
英文摘要
Fourier-based methods are among the most widely used techniques for pricing European options when the characteristic function of the underlying asset process is available. Their applicability to increasingly fine discretizations, however, is limited by the rapidly growing memory requirements of classical Fourier transforms, which become a computational bottleneck for large-scale pricing problems. In this work, we overcome this limitation by reformulating the Carr-Madan pricing framework using tensor networks. Specifically, we employ the Superfast Fourier Transform (SFFT), a compressed Tensor Train representation of the Quantum Fourier Transform (QFT), and apply it directly to tensorized option pricing without ever explicitly constructing exponentially large vectors or Fourier operators. This formulation also enables a direct comparison between the classical tensor network algorithm and its quantum counterpart through QFT-based option pricing on quantum simulators and quantum hardware. Numerical experiments for European call options demonstrate that the proposed SFFT method maintains pricing accuracy while substantially reducing memory requirements and achieving subexponential computational scaling compared with conventional FFT-based pricing. The accompanying quantum simulations and hardware executions enable a direct comparison between the classical tensor network formulation and its QFT-based quantum counterpart, showing that both approaches avoid the exponential scaling of conventional Fourier implementations and provide complementary perspectives on large-scale option pricing. Together, these results establish a unified framework connecting classical Fourier pricing, tensor network algorithms, and quantum computing approaches, demonstrating how tensorized Fourier methods can provide scalable alternatives for high-dimensional financial computations.
