求解金融中高维期权定价 Black-Scholes 偏微分方程的量子蒙特卡洛算法及其复杂度分析
Quantum Monte Carlo algorithm for solving Black-Scholes PDEs for high-dimensional option pricing in finance and its complexity analysis
AI总结：
本文提出一种量子蒙特卡洛算法求解带相关性的高维 Black-Scholes PDE 以进行期权定价，证明了其在空间维度和精度倒数上具有多项式复杂度界，并在有界收益函数下相比经典方法实现加速。
摘要：
在本文中，我们提供了一种量子蒙特卡洛算法，用于求解带相关性的高维 Black-Scholes PDE，以进行高维期权定价。期权的收益函数为一般形式，仅要求其连续且分段仿射（CPWA），这涵盖了金融中使用的大多数相关收益函数。我们对算法进行了严格的误差分析和复杂度分析。特别地，我们证明了算法的计算复杂度在 PDE 的空间维度 $d$ 和预设精度 $\varepsilon$ 的倒数上受多项式界限。此外，我们表明对于有界的收益函数，与经典蒙特卡洛方法相比，我们的算法确实实现了加速。进一步地，我们使用在 Qiskit 框架内开发的包，在 1 维和 2 维中进行了数值模拟，该包专为针对 Black-Scholes 模型对 CPWA 期权定价而定制，并讨论了将数值模拟推广至任意空间维度的潜在扩展。
英文摘要：
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.
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