Quantum algorithms for the pricing of financial derivatives have been discussed in recent papers. However, the pricing model discussed in those papers is too simple for practical purposes. It motivates us to consider how to implement more complex models used in financial institutions. In this paper, we consider the local volatility (LV) model, in which the volatility of the underlying asset price depends on the price and time. As in previous studies, we use the quantum amplitude estimation (QAE) as the main source of quantum speedup and discuss the state preparation step of the QAE, or equivalently, the implementation of the asset price evolution. We compare two types of state preparation: One is the amplitude encoding (AE) type, where the probability distribution of the derivative’s payoff is encoded to the probabilistic amplitude. The other is the pseudo-random number (PRN) type, where sequences of PRNs are used to simulate the asset price evolution as in classical Monte Carlo simulation. We present detailed circuit diagrams for implementing these preparation methods in fault-tolerant quantum computation and roughly estimate required resources such as the number of qubits and T-count.

中文翻译：

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微笑的量子定价：在量子计算机上实现局部波动率模型

在最近的论文中讨论了用于金融衍生品定价的量子算法。然而，这些论文中讨论的定价模型对于实际目的来说太简单了。它促使我们考虑如何实现金融机构中使用的更复杂的模型。在本文中，我们考虑局部波动率（LV）模型，其中标的资产价格的波动率取决于价格和时间。与之前的研究一样，我们使用量子幅度估计（QAE）作为量子加速的主要来源，并讨论了 QAE 的状态准备步骤，或者等效地，资产价格演化的实现。我们比较了两种类型的状态准备：一种是幅度编码（AE）类型，其中导数收益的概率分布被编码为概率幅度。另一种是伪随机数 (PRN) 类型，其中 PRN 序列用于模拟资产价格演变，就像在经典蒙特卡洛模拟中一样。我们提供了在容错量子计算中实现这些制备方法的详细电路图，并粗略估计了所需的资源，例如量子比特数和 T 计数。