  Inspired by recent progress in quantum algorithms for ordinary and partial
differential equations, we study quantum algorithms for stochastic differential
equations (SDEs). Firstly we provide a quantum algorithm that gives a quadratic
speed-up for multilevel Monte Carlo methods in a general setting. As
applications, we apply it to compute expectation values determined by classical
solutions of SDEs, with improved dependence on precision. We demonstrate the
use of this algorithm in a variety of applications arising in mathematical
finance, such as the Black-Scholes and Local Volatility models, and Greeks. We
also provide a quantum algorithm based on sublinear binomial sampling for the
binomial option pricing model with the same improvement.
