  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.
