This paper explores advancements in quantum algorithms for derivative pricing of exotics, a computational pipeline of fundamental importance in quantitative finance. For such cases, the classical Monte Carlo integration procedure provides the state-of-the-art provable, asymptotic performance: polynomial in problem dimension and quadratic in inverse-precision. While quantum algorithms are known to offer quadratic speedups over classical Monte Carlo methods, end-to-end speedups have been proven only in the simplified setting over the Black-Scholes geometric Brownian motion (GBM) model. This paper extends existing frameworks to demonstrate novel quadratic speedups for more practical models, such as the Cox-Ingersoll-Ross (CIR) model and a variant of Heston's stochastic volatility model, utilizing a characteristic of the underlying SDEs which we term fast-forwardability. Additionally, for general models that do not possess the fast-forwardable property, we introduce a quantum Milstein sampler, based on a novel quantum algorithm for sampling Lévy areas, which enables quantum multi-level Monte Carlo to achieve quadratic speedups for multi-dimensional stochastic processes exhibiting certain correlation types. We also present an improved analysis of numerical integration for derivative pricing, leading to substantial reductions in the resource requirements for pricing GBM and CIR models. Furthermore, we investigate the potential for additional reductions using arithmetic-free quantum procedures. Finally, we critique quantum partial differential equation (PDE) solvers as a method for derivative pricing based on amplitude estimation, identifying theoretical barriers that obstruct achieving a quantum speedup through this approach. Our findings significantly advance the understanding of quantum algorithms in derivative pricing, addressing key challenges and open questions in the field.

翻译：本文探讨了奇异衍生品定价量子算法的进展，这是量化金融中一个具有根本重要性的计算流程。对于此类情况，经典的蒙特卡洛积分方法提供了最先进的可证明渐近性能：在问题维度上为多项式复杂度，在逆精度上为二次复杂度。虽然已知量子算法相较于经典蒙特卡洛方法能提供二次加速，但端到端的加速仅在Black-Scholes几何布朗运动（GBM）模型的简化设定下得到了证明。本文扩展了现有框架，针对更实际的模型（如Cox-Ingersoll-Ross（CIR）模型和Heston随机波动率模型的一个变体）展示了新颖的二次加速，其利用了底层随机微分方程的一个我们称之为快速前推性的特性。此外，对于不具备快速前推性的一般模型，我们引入了一种基于新颖的Lévy面积采样量子算法的量子Milstein采样器，该采样器使得量子多级蒙特卡洛方法能够对呈现特定相关类型的多维随机过程实现二次加速。我们还提出了对衍生品定价数值积分的改进分析，从而大幅降低了GBM和CIR模型定价的资源需求。此外，我们研究了使用无算术量子流程实现进一步资源缩减的潜力。最后，我们批判性地分析了基于振幅估计的量子偏微分方程（PDE）求解器作为衍生品定价方法的可行性，指出了阻碍通过此途径实现量子加速的理论障碍。我们的研究结果显著推进了对衍生品定价量子算法的理解，解决了该领域的关键挑战和开放性问题。