Pricing multi-asset derivatives by finite difference method on a quantum computer

Following the recent great advance of quantum computing technology, there are growing interests in its applications to industries, including finance. In this paper, we focus on derivative pricing based on solving the Black-Scholes partial differential equation by finite difference method (FDM), which is a suitable approach for some types of derivatives but suffers from the {\it curse of dimensionality}, that is, exponential growth of complexity in the case of multiple underlying assets. We propose a quantum algorithm for FDM-based pricing of multi-asset derivative with exponential speedup with respect to dimensionality compared with classical algorithms. The proposed algorithm utilizes the quantum algorithm for solving differential equations, which is based on quantum linear system algorithms. Addressing the specific issue in derivative pricing, that is, extracting the derivative price for the present underlying asset prices from the output state of the quantum algorithm, we present the whole of the calculation process and estimate its complexity. We believe that the proposed method opens the new possibility of accurate and high-speed derivative pricing by quantum computers.