In this paper we provide a quantum Monte Carlo algorithm to solve high-dimensional Black-Scholes PDEs with correlation for high-dimensional option pricing. The payoff function of the option is of general form and is only required to be continuous and piece-wise affine (CPWA), which covers most of the relevant payoff functions used in finance. We provide a rigorous error analysis and complexity analysis of our algorithm. In particular, we prove that the computational complexity of our algorithm is bounded polynomially in the space dimension $d$ of the PDE and the reciprocal of the prescribed accuracy $\varepsilon$. Moreover, we show that for payoff functions which are bounded, our algorithm indeed has a speed-up compared to classical Monte Carlo methods. Furthermore, we provide numerical simulations in one and two dimensions using our developed package within the Qiskit framework tailored to price CPWA options with respect to the Black-Scholes model, as well as discuss the potential extension of the numerical simulations to arbitrary space dimension.

翻译：本文提出一种量子蒙特卡洛算法，用于求解含相关性的高维Black-Scholes偏微分方程，以实现高维期权定价。期权的收益函数为一般形式，仅需满足连续且分段仿射（CPWA）条件，这涵盖了金融领域绝大多数相关收益函数。我们对该算法进行了严格的误差分析和复杂度分析，特别证明了算法计算复杂度关于偏微分方程空间维度$d$及预设精度$\varepsilon$的倒数呈多项式有界性。此外，对于有界收益函数，我们的算法相比经典蒙特卡洛方法确实具有加速效果。进一步地，我们利用基于Qiskit框架自主开发的软件包，在一维和二维空间进行了针对Black-Scholes模型下CPWA期权定价的数值模拟，并探讨了将数值模拟扩展至任意空间维度的潜在可能性。