Pricing interest-rate financial derivatives is a major problem in finance, in which it is crucial to accurately reproduce the time evolution of interest rates. Several stochastic dynamics have been proposed in the literature to model either the instantaneous interest rate or the instantaneous forward rate. A successful approach to model the latter is the celebrated Heath-Jarrow-Morton framework, in which its dynamics is entirely specified by volatility factors. In its multifactor version, this model considers several noisy components to capture at best the dynamics of several time-maturing forward rates. However, as no general analytical solution is available, there is a trade-off between the number of noisy factors considered and the computational time to perform a numerical simulation. Here, we employ the quantum principal component analysis to reduce the number of noisy factors required to accurately simulate the time evolution of several time-maturing forward rates. The principal components are experimentally estimated with the five-qubit IBMQX2 quantum computer for $2\times 2$ and $3\times 3$ cross-correlation matrices, which are based on historical data for two and three time-maturing forward rates. This paper is a step towards the design of a general quantum algorithm to fully simulate on quantum computers the Heath-Jarrow-Morton model for pricing interest-rate financial derivatives. It shows indeed that practical applications of quantum computers in finance will be achievable in the near future.

中文翻译：

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使用IBM量子计算机对金融衍生产品进行定价

利率金融衍生产品的定价是金融中的一个主要问题，其中准确再现利率的时间变化至关重要。在文献中已经提出了几种随机动力学来对瞬时利率或瞬时远期利率建模。成功的后者建模方法是著名的Heath-Jarrow-Morton框架，该框架的动力学完全由波动性因素确定。在其多因素版本中，该模型考虑了多个噪声成分，以最大程度地捕获多个时间成熟的远期利率的动态变化。但是，由于没有通用的分析解决方案，因此在考虑的噪声因素数量与执行数值模拟的计算时间之间需要进行权衡。这里，我们使用量子主成分分析来减少为精确模拟几种时间到期的远期利率的时间演变所需的噪声因子的数量。使用五量子位的IBMQX2量子计算机对主要成分进行了实验估算，$2\times 2$ 和 $3\times 3$互相关矩阵，它们基于两个和三个时间到期的远期利率的历史数据。本文是迈向设计通用量子算法的一步，该算法将在量子计算机上完全模拟Heath-Jarrow-Morton模型以对利率金融衍生产品定价。它确实表明，在不久的将来将可以实现量子计算机在金融中的实际应用。