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

Dong An; Noah Linden; Jin-Peng Liu; Ashley Montanaro; Changpeng Shao; Jiasu Wang

doi:10.22331/q-2021-06-24-481

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

Dong An¹, Noah Linden², Jin-Peng Liu^3,4,5, Ashley Montanaro^2,6, 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

Published:	2021-06-24, volume 5, page 481
Eprint:	arXiv:2012.06283v2
Doi:	https://doi.org/10.22331/q-2021-06-24-481
Citation:	Quantum 5, 481 (2021).

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Abstract

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.

► BibTeX data

@article{An2021quantumaccelerated,
  doi = {10.22331/q-2021-06-24-481},
  url = {https://doi.org/10.22331/q-2021-06-24-481},
  title = {Quantum-accelerated multilevel {M}onte {C}arlo methods for stochastic differential equations in mathematical finance},
  author = {An, Dong and Linden, Noah and Liu, Jin-Peng and Montanaro, Ashley and Shao, Changpeng and Wang, Jiasu},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {481},
  month = jun,
  year = {2021}
}

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