We develop quantum algorithms for pricing Asian and barrier options under the Heston model, a popular stochastic volatility model, and estimate their costs, in terms of T-count, T-depth and number of logical qubits, on instances under typical market conditions. These algorithms are based on combining well-established numerical methods for stochastic differential equations and quantum amplitude estimation technique. In particular, we empirically show that, despite its simplicity, weak Euler method achieves the same level of accuracy as the better-known strong Euler method in this task. Furthermore, by eliminating the expensive procedure of preparing Gaussian states, the quantum algorithm based on weak Euler scheme achieves drastically better efficiency than the one based on strong Euler scheme. Our resource analysis suggests that option pricing under stochastic volatility is a promising application of quantum computers, and that our algorithms render the hardware requirement for reaching practical quantum advantage in financial applications less stringent than prior art.

中文翻译：

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量子计算机上随机波动率下的期权定价


我们开发了量子算法，用于在 Heston 模型（一种流行的随机波动率模型）下为亚洲期权和障碍期权定价，并根据典型市场条件下实例的 T 计数、T 深度和逻辑量子比特的数量来估计它们的成本。这些算法基于随机微分方程的成熟数值方法和量子振幅估计技术的结合。特别是，我们实证表明，尽管它很简单，但在这项任务中，弱欧拉方法达到了与更广为人知的强欧拉方法相同的精度水平。此外，由于省去了准备高斯态的昂贵过程，基于弱欧拉方案的量子算法比基于强欧拉方案的量子算法效率高得多。我们的资源分析表明，随机波动率下的期权定价是量子计算机的一个有前途的应用，并且我们的算法使得在金融应用中实现实际量子优势的硬件要求不如现有技术严格。