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：本文提出了一种量子蒙特卡洛算法，用于求解具有相关性的高维Black-Scholes偏微分方程，以进行高维期权定价。该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，这涵盖了金融领域中使用的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性分析。特别地，我们证明了算法的计算复杂度在偏微分方程的空间维度$d$和预设精度$\varepsilon$的倒数上呈多项式有界。此外，我们表明对于有界的收益函数，该算法相较于经典蒙特卡洛方法确实具有加速优势。进一步地，我们利用在Qiskit框架内开发的专用软件包，针对Black-Scholes模型对CPWA期权进行了一维和二维的数值模拟，并讨论了将数值模拟扩展至任意空间维度的潜在可能性。