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

Research output: Contribution to journalArticle (Academic Journal)peer-review

29 Citations (Scopus)

Abstract

Inspired by recent progress in quantum algorithms for ordinary and partial differential equations, we study quantum algorithms for stochastic differen- tial 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.

Original languageEnglish
JournalQuantum
Volume5
DOIs
Publication statusPublished - 24 Jun 2021

Bibliographical note

Funding Information:
The authors thank Andrew M. Childs, Lin Lin, and Nick Whiteley for valuable discussions and comments. JPL did part of this work while visiting the Simons Institute for the Theory of Computing in Berkeley and gratefully acknowledge its hospitality. We thank the National Energy Research Scientific Computing (NERSC) center and the Berkeley Research Computing (BRC) program at the University of California, Berkeley for making computational resources available. The authors acknowledge support from National Science Foundation grant CCF-1813814, the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Quantum Algorithms Teams and Accelerated Research in Quantum Computing programs. This work was also partially supported by the Department of Energy un- der Grant No. DE-SC0017867 and the National Science Foundation under the Quantum Leap Challenge Institutes (D.A.,J.W.). We acknowledge support from the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union's Horizon 2020 Programme (QuantAlgo project) and EPSRC grants EP/R043957/1 and EP/T001062/1. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 817581).

Funding Information:
The authors acknowledge support from National Science Foundation grant CCF-1813814, the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Quantum Algorithms Teams and Accelerated Research in Quantum Computing programs. This work was also partially supported by the Department of Energy under Grant No. DE-SC0017867 and the National Science Foundation under the Quantum Leap Challenge Institutes (D.A.,J.W.). We acknowledge support from the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union’s Horizon 2020 Programme (QuantAlgo project) and EPSRC grants EP/R043957/1 and EP/T001062/1. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 817581).

Publisher Copyright:
© 2021 Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften.

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