Pricing multi-asset derivatives by variational quantum algorithms

TL;DR

This paper introduces a variational quantum algorithm for multi-asset derivative pricing that reduces computational complexity and measurement bottlenecks, making quantum approaches feasible on small-scale noisy quantum computers.

Contribution

It proposes a novel variational quantum simulation method to solve the Black-Scholes equation for derivative pricing, overcoming measurement bottlenecks and enabling practical quantum computations.

Findings

Successfully simulated derivative pricing using small quantum computers.

Demonstrated potential quantum speedup in derivative valuation.

Validated the approach with numerical experiments.

Abstract

Pricing a multi-asset derivative is an important problem in financial engineering, both theoretically and practically. Although it is suitable to numerically solve partial differential equations to calculate the prices of certain types of derivatives, the computational complexity increases exponentially as the number of underlying assets increases in some classical methods, such as the finite difference method. Therefore, there are efforts to reduce the computational complexity by using quantum computation. However, when solving with naive quantum algorithms, the target derivative price is embedded in the amplitude of one basis of the quantum state, and so an exponential complexity is required to obtain the solution. To avoid the bottleneck, the previous study~[Miyamoto and Kubo, IEEE Transactions on Quantum Engineering, \textbf{3}, 1--25 (2022)] utilizes the fact that the present price of a derivative can be obtained by its discounted expected value at any future point in time and shows that the quantum algorithm can reduce the complexity. In this paper, to make the algorithm feasible to run on a small quantum computer, we use variational quantum simulation to solve the Black-Scholes equation and compute the derivative price from the inner product between the solution and a probability distribution. This avoids the measurement bottleneck of the naive approach and would provide quantum speedup even in noisy quantum computers. We also conduct numerical experiments to validate our method. Our method will be an important breakthrough in derivative pricing using small-scale quantum computers.