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Quantum Monte Carlo algorithm for solving Black-Scholes PDEs for high-dimensional option pricing in finance and its complexity analysis

Author

Listed:
  • Jianjun Chen
  • Yongming Li
  • Ariel Neufeld

Abstract

In this paper we provide a quantum Monte Carlo algorithm to solve high-dimensional Black-Scholes PDEs with correlation for high-dimensional option pricing. The payoff function of the option is of general form and is only required to be continuous and piece-wise affine (CPWA), which covers most of the relevant payoff functions used in finance. We provide a rigorous error analysis and complexity analysis of our algorithm. In particular, we prove that the computational complexity of our algorithm is bounded polynomially in the space dimension $d$ of the PDE and the reciprocal of the prescribed accuracy $\varepsilon$. Moreover, we show that for payoff functions which are bounded, our algorithm indeed has a speed-up compared to classical Monte Carlo methods. Furthermore, we provide numerical simulations in one and two dimensions using our developed package within the Qiskit framework tailored to price CPWA options with respect to the Black-Scholes model, as well as discuss the potential extension of the numerical simulations to arbitrary space dimension.

Suggested Citation

  • Jianjun Chen & Yongming Li & Ariel Neufeld, 2023. "Quantum Monte Carlo algorithm for solving Black-Scholes PDEs for high-dimensional option pricing in finance and its complexity analysis," Papers 2301.09241, arXiv.org, revised Apr 2024.
    • Handle: RePEc:arx:papers:2301.09241
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    References listed on IDEAS

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    1. 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.
    2. Dong An & Noah Linden & Jin-Peng Liu & Ashley Montanaro & Changpeng Shao & Jiasu Wang, 2020. "Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance," Papers 2012.06283, arXiv.org, revised Jun 2021.
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    Cited by:

    1. Antoine Jacquier & Oleksiy Kondratyev & Gordon Lee & Mugad Oumgari, 2023. "Quantum Computing for Financial Mathematics," Papers 2311.06621, arXiv.org.
    2. Raj G. Patel & Tomas Dominguez & Mohammad Dib & Samuel Palmer & Andrea Cadarso & Fernando De Lope Contreras & Abdelkader Ratnani & Francisco Gomez Casanova & Senaida Hern'andez-Santana & 'Alvaro D'iaz, 2023. "Application of Tensor Neural Networks to Pricing Bermudan Swaptions," Papers 2304.09750, arXiv.org, revised Mar 2024.
    3. Ariel Neufeld & Julian Sester, 2023. "Neural networks can detect model-free static arbitrage strategies," Papers 2306.16422, arXiv.org, revised Aug 2024.

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