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Quantum algorithm for stochastic optimal stopping problems with applications in finance

Author

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  • Jo~ao F. Doriguello
  • Alessandro Luongo
  • Jinge Bao
  • Patrick Rebentrost
  • Miklos Santha

Abstract

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.

Suggested Citation

  • Jo~ao F. Doriguello & Alessandro Luongo & Jinge Bao & Patrick Rebentrost & Miklos Santha, 2021. "Quantum algorithm for stochastic optimal stopping problems with applications in finance," Papers 2111.15332, arXiv.org, revised Jul 2023.
    • Handle: RePEc:arx:papers:2111.15332
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    References listed on IDEAS

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    1. Kazuya Kaneko & Koichi Miyamoto & Naoyuki Takeda & Kazuyoshi Yoshino, 2020. "Quantum Pricing with a Smile: Implementation of Local Volatility Model on Quantum Computer," Papers 2007.01467, arXiv.org.
    2. Daniel Z. Zanger, 2018. "Convergence Of A Least†Squares Monte Carlo Algorithm For American Option Pricing With Dependent Sample Data," Mathematical Finance, Wiley Blackwell, vol. 28(1), pages 447-479, January.
    3. Lars Stentoft, 2004. "Convergence of the Least Squares Monte Carlo Approach to American Option Valuation," Management Science, INFORMS, vol. 50(9), pages 1193-1203, September.
    4. Powell, Warren B., 2019. "A unified framework for stochastic optimization," European Journal of Operational Research, Elsevier, vol. 275(3), pages 795-821.
    5. Philip Protter & Emmanuelle Clément & Damien Lamberton, 2002. "An analysis of a least squares regression method for American option pricing," Finance and Stochastics, Springer, vol. 6(4), pages 449-471.
    6. Shouvanik Chakrabarti & Rajiv Krishnakumar & Guglielmo Mazzola & Nikitas Stamatopoulos & Stefan Woerner & William J. Zeng, 2020. "A Threshold for Quantum Advantage in Derivative Pricing," Papers 2012.03819, arXiv.org, revised May 2021.
    7. Daniel Zanger, 2009. "Convergence of a Least-Squares Monte Carlo Algorithm for Bounded Approximating Sets," Applied Mathematical Finance, Taylor & Francis Journals, vol. 16(2), pages 123-150.
    8. Anne-Sophie Krah & Zoran Nikolić & Ralf Korn, 2018. "A Least-Squares Monte Carlo Framework in Proxy Modeling of Life Insurance Companies," Risks, MDPI, vol. 6(2), pages 1-26, June.
    9. Chen Liu & Henry Schellhorn & Qidi Peng, 2019. "American Option Pricing With Regression: Convergence Analysis," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(08), pages 1-31, December.
    10. Lars Stentoft, 2004. "Assessing the Least Squares Monte-Carlo Approach to American Option Valuation," Review of Derivatives Research, Springer, vol. 7(2), pages 129-168, August.
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    Cited by:

    1. Jeong Yu Han & Patrick Rebentrost, 2022. "Quantum advantage for multi-option portfolio pricing and valuation adjustments," Papers 2203.04924, arXiv.org.

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