Quantum Algorithm for Stochastic Optimal Stopping Problems with Applications in Finance

Authors João F. Doriguello , Alessandro Luongo, Jinge Bao, Patrick Rebentrost, Miklos Santha

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Author Details

João F. Doriguello
  • Centre for Quantum Technologies, National University of Singapore, Singapore
Alessandro Luongo
  • Centre for Quantum Technologies, National University of Singapore, Singapore
Jinge Bao
  • Centre for Quantum Technologies, National University of Singapore, Singapore
Patrick Rebentrost
  • Centre for Quantum Technologies, National University of Singapore, Singapore
Miklos Santha
  • Centre for Quantum Technologies, National University of Singapore, Singapore


We thank Rajagopal Raman for pointing out Ref. [Krah et al., 2018] and the importance of the Longstaff-Schwarz algorithm in the insurance industry. We also thank Koichi Miyamoto for Ref. [Kaneko et al., 2021] and Eric Ghysels for Refs. [Broadie et al., 2000; Broadie et al., 2000].

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João F. Doriguello, Alessandro Luongo, Jinge Bao, Patrick Rebentrost, and Miklos Santha. Quantum Algorithm for Stochastic Optimal Stopping Problems with Applications in Finance. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 2:1-2:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


The famous least squares Monte Carlo (LSM) algorithm combines linear least square regression with Monte Carlo simulation to approximately solve problems in stochastic optimal stopping theory. In this work, we propose a quantum LSM based on quantum access to a stochastic process, on quantum circuits for computing the optimal stopping times, and on quantum techniques for Monte Carlo. For this algorithm, we elucidate the intricate interplay of function approximation and quantum algorithms for Monte Carlo. Our algorithm achieves a nearly quadratic speedup in the runtime compared to the LSM algorithm under some mild assumptions. Specifically, our quantum algorithm can be applied to American option pricing and we analyze a case study for the common situation of Brownian motion and geometric Brownian motion processes.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Stochastic processes
  • Mathematics of computing → Markov-chain Monte Carlo methods
  • Theory of computation → Quantum computation theory
  • Quantum computation complexity
  • optimal stopping time
  • stochastic processes
  • American options
  • quantum finance


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