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Closed Quantum Black-Scholes: Quantum Drift and the Heisenberg Equation of Motion

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  • Will Hicks

Abstract

In this article we model a financial derivative price as an observable on the market state function. We apply geometric techniques to integrating the Heisenberg Equation of Motion. We illustrate how the non-commutative nature of the model introduces quantum interference effects that can act as either a drag or a boost on the resulting return. The ultimate objective is to investigate the nature of quantum drift in the Accardi-Boukas quantum Black-Scholes framework which involves modelling the financial market as a quantum observable, and introduces randomness through the Hudson-Parthasarathy quantum stochastic calculus. In particular we aim to differentiate randomness that is introduced through external noise (quantum stochastic calculus) and randomness that is fundamental to a quantum system (Heisenberg Equation of Motion).

Suggested Citation

  • Will Hicks, 2019. "Closed Quantum Black-Scholes: Quantum Drift and the Heisenberg Equation of Motion," Papers 1911.11475, arXiv.org, revised Jan 2020.
    • Handle: RePEc:arx:papers:1911.11475
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    References listed on IDEAS

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    1. Piotrowski, E.W & Sładkowski, J, 2002. "Quantum market games," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 312(1), pages 208-216.
    2. Will Hicks, 2018. "PT Symmetry, Non-Gaussian Path Integrals, and the Quantum Black-Scholes Equation," Papers 1812.00839, arXiv.org, revised Jan 2019.
    3. Piotrowski, Edward W. & Sładkowski, Jan, 2005. "Quantum diffusion of prices and profits," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 345(1), pages 185-195.
    4. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    5. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    6. Will Hicks, 2019. "A Nonlocal Approach to The Quantum Kolmogorov Backward Equation and Links to Noncommutative Geometry," Papers 1905.07257, arXiv.org.
    7. Will Hicks, 2018. "Nonlocal Diffusions and The Quantum Black-Scholes Equation: Modelling the Market Fear Factor," Papers 1806.07983, arXiv.org, revised Jun 2018.
    8. Edward W. Piotrowski & Jan Sladkowski, "undated". "Quantum-Like Approach to Financial Risk: Quantum Anthropic Principle," Departmental Working Papers 8, University of Bialtystok, Department of Theoretical Physics.
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    Cited by:

    1. Anantya Bhatnagar & Dimitri D. Vvedensky, 2022. "Quantum effects in an expanded Black–Scholes model," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 95(8), pages 1-12, August.
    2. Yeşiltaş, Özlem, 2023. "The Black–Scholes equation in finance: Quantum mechanical approaches," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 623(C).
    3. Will Hicks, 2020. "Pseudo-Hermiticity, Martingale Processes and Non-Arbitrage Pricing," Papers 2009.00360, arXiv.org, revised Apr 2021.
    4. Will Hicks, 2021. "Wild Randomness, and the application of Hyperbolic Diffusion in Financial Modelling," Papers 2101.04604, arXiv.org, revised Apr 2021.
    5. Will Hicks, 2023. "Modelling Illiquid Stocks Using Quantum Stochastic Calculus," Papers 2302.05243, arXiv.org.

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