Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance

Dong An1, Noah Linden2, Jin-Peng Liu3,4,5, Ashley Montanaro2,6, Changpeng Shao2, and Jiasu Wang1

1Department of Mathematics, University of California, Berkeley, CA 94720, USA
2School of Mathematics, Fry Building, University of Bristol, BS8 1UG, UK
3Joint Center for Quantum Information and Computer Science, University of Maryland, MD 20742, USA
4Institute for Advanced Computer Studies, University of Maryland, MD 20742, USA
5Department of Mathematics, University of Maryland, MD 20742, USA
6Phasecraft Ltd, Quantum Technologies Innovation Centre, Bristol BS1 5DD, UK

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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.

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