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Pseudo-Hermiticity, Martingale Processes and Non-Arbitrage Pricing

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  • Will Hicks

Abstract

Financial models based on the Wick product, and White Noise formalism have previously been suggested in order to incorporate integrals with respect to fractional Brownian motion. It has also been pointed out that this leads naturally to a quantum mechanical interpretation of the financial market. In this article we pursue this idea further, and in particular show how the framework of quantum probability can be used to construct Martingales, without relying on Brownian integrals. We go on to suggest benefits of doing so, and avenues for future work.

Suggested Citation

  • Will Hicks, 2020. "Pseudo-Hermiticity, Martingale Processes and Non-Arbitrage Pricing," Papers 2009.00360, arXiv.org, revised Apr 2021.
    • Handle: RePEc:arx:papers:2009.00360
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    References listed on IDEAS

    as
    1. Will Hicks, 2018. "PT Symmetry, Non-Gaussian Path Integrals, and the Quantum Black-Scholes Equation," Papers 1812.00839, arXiv.org, revised Jan 2019.
    2. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742.
    3. Contreras, Mauricio & Pellicer, Rely & Villena, Marcelo & Ruiz, Aaron, 2010. "A quantum model of option pricing: When Black–Scholes meets Schrödinger and its semi-classical limit," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(23), pages 5447-5459.
    4. Haven, Emmanuel, 2003. "A Black-Scholes Schrödinger option price: ‘bit’ versus ‘qubit’," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(1), pages 201-206.
    5. Will Hicks, 2019. "A Nonlocal Approach to The Quantum Kolmogorov Backward Equation and Links to Noncommutative Geometry," Papers 1905.07257, arXiv.org.
    6. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305, World Scientific Publishing Co. Pte. Ltd..
    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. Will Hicks, 2019. "Closed Quantum Black-Scholes: Quantum Drift and the Heisenberg Equation of Motion," Papers 1911.11475, arXiv.org, revised Jan 2020.
    9. Jana, T.K. & Roy, P., 2012. "Pseudo Hermitian formulation of the quantum Black–Scholes Hamiltonian," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(8), pages 2636-2640.
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