基于量子振幅估计和Chebyshev插值方法的期权定价
Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation
摘要：
金融衍生品的定价，特别是可提前行使的期权，如期权，是金融机构的一项重要但繁重的数字任务，其速度将产生巨大的业务影响。最近，已经开始研究量子计算在金融问题上的应用。在本文中，我们首先提出了一个量子算法来计算期权定价。该方法通过Chebyshev插值，利用量子振幅估计方法估计插值节点处的值，对连续值进行逼近，这是期权定价的关键部分。在此方法中，调用oracle生成标的资产价格路径的次数按比例缩放为$\widetilde{O}(ε^{-1})$，其中$ε$是期权价格的误差容限。这意味着与经典的基于蒙特卡罗的方法（例如最小二乘蒙特卡罗，其中Oracle调用号是$\widetilde{O}(ε^{-2})$）相比，速度提高了二倍。
英文摘要：
Pricing of financial derivatives, in particular early exercisable options such as Bermudan options, is an important but heavy numerical task in financial institutions, and its speed-up will provide a large business impact. Recently, applications of quantum computing to financial problems have been started to be investigated. In this paper, we first propose a quantum algorithm for Bermudan option pricing. This method performs the approximation of the continuation value, which is a crucial part of Bermudan option pricing, by Chebyshev interpolation, using the values at interpolation nodes estimated by quantum amplitude estimation. In this method, the number of calls to the oracle to generate underlying asset price paths scales as $\widetilde{O}(ε^{-1})$, where $ε$ is the error tolerance of the option price. This means the quadratic speed-up compared with classical Monte Carlo-based methods such as least-squares Monte Carlo, in which the oracle call number is $\widetilde{O}(ε^{-2})$.
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