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A quantum model of option pricing: When Black–Scholes meets Schrödinger and its semi-classical limit

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  • Contreras, Mauricio
  • Pellicer, Rely
  • Villena, Marcelo
  • Ruiz, Aaron

Abstract

The Black–Scholes equation can be interpreted from the point of view of quantum mechanics, as the imaginary time Schrödinger equation of a free particle. When deviations of this state of equilibrium are considered, as a product of some market imperfection, such as: Transaction cost, asymmetric information issues, short-term volatility, extreme discontinuities, or serial correlations; the classical non-arbitrage assumption of the Black–Scholes model is violated, implying a non-risk-free portfolio. From Haven (2002) [1] we know that an arbitrage environment is a necessary condition to embedding the Black–Scholes option pricing model in a more general quantum physics setting. The aim of this paper is to propose a new Black–Scholes–Schrödinger model based on the endogenous arbitrage option pricing formulation introduced by Contreras et al. (2010) [2]. Hence, we derive a more general quantum model of option pricing, that incorporates arbitrage as an external time dependent force, which has an associated potential related to the random dynamic of the underlying asset price. This new resultant model can be interpreted as a Schrödinger equation in imaginary time for a particle of mass 1/σ2 with a wave function in an external field force generated by the arbitrage potential. As pointed out above, this new model can be seen as a more general formulation, where the perfect market equilibrium state postulated by the Black–Scholes model represent a particular case. Finally, since the Schrödinger equation is in place, we can apply semiclassical methods, of common use in theoretical physics, to find an approximate analytical solution of the Black–Scholes equation in the presence of market imperfections, as it is the case of an arbitrage bubble. Here, as a numerical illustration of the potential of this Schrödinger equation analogy, the semiclassical approximation is performed for different arbitrage bubble forms (step, linear and parabolic) and compare with the exact solution of our general quantum model of option pricing.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:389:y:2010:i:23:p:5447-5459
    DOI: 10.1016/j.physa.2010.08.018
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    References listed on IDEAS

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    Cited by:

    1. Kartono, Agus & Solekha, Siti & Sumaryada, Tony & Irmansyah,, 2021. "Foreign currency exchange rate prediction using non-linear Schrödinger equations with economic fundamental parameters," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    2. Will Hicks, 2020. "Pseudo-Hermiticity, Martingale Processes and Non-Arbitrage Pricing," Papers 2009.00360, arXiv.org, revised Apr 2021.
    3. Axel A. Araneda & Marcelo J. Villena, 2018. "Computing the CEV option pricing formula using the semiclassical approximation of path integral," Papers 1803.10376, arXiv.org.
    4. Mauricio Contreras G, 2020. "An Application of Dirac's Interaction Picture to Option Pricing," Papers 2010.06747, arXiv.org.
    5. G., Mauricio Contreras & Peña, Juan Pablo, 2019. "The quantum dark side of the optimal control theory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 515(C), pages 450-473.
    6. Contreras, M. & Echeverría, J. & Peña, J.P. & Villena, M., 2020. "Resonance phenomena in option pricing with arbitrage," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    7. Contreras, Mauricio & Pellicer, Rely & Villena, Marcelo, 2017. "Dynamic optimization and its relation to classical and quantum constrained systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 479(C), pages 12-25.
    8. Jena, Rajarama Mohan & Chakraverty, Snehashish & Baleanu, Dumitru, 2020. "A novel analytical technique for the solution of time-fractional Ivancevic option pricing model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 550(C).
    9. Denis M. Filatov & Maksim A. Vanyarkho, 2014. "An Unconventional Attempt to Tame Mandelbrot's Grey Swans," Papers 1406.5718, arXiv.org.
    10. Chowdhury, Reaz & Mahdy, M.R.C. & Alam, Tanisha Nourin & Al Quaderi, Golam Dastegir & Arifur Rahman, M., 2020. "Predicting the stock price of frontier markets using machine learning and modified Black–Scholes Option pricing model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 555(C).
    11. Bustamante, M. & Contreras, M., 2016. "Multi-asset Black–Scholes model as a variable second class constrained dynamical system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 457(C), pages 540-572.
    12. Mauricio Contreras G, 2020. "Endogenous Stochastic Arbitrage Bubbles and the Black--Scholes model," Papers 2009.09329, arXiv.org.
    13. 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.
    14. Mauricio Contreras & Rely Pellicer & Daniel Santiagos & Marcelo Villena, 2015. "Calibration and simulation of arbitrage effects in a non-equilibrium quantum Black-Scholes model by using semiclassical methods," Papers 1512.05377, arXiv.org.
    15. Contreras, Mauricio & Hojman, Sergio A., 2014. "Option pricing, stochastic volatility, singular dynamics and constrained path integrals," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 393(C), pages 391-403.
    16. Reaz Chowdhury & M. R. C. Mahdy & Tanisha Nourin Alam & Golam Dastegir Al Quaderi, 2018. "Predicting the Stock Price of Frontier Markets Using Modified Black-Scholes Option Pricing Model and Machine Learning," Papers 1812.10619, arXiv.org.
    17. Contreras G., Mauricio, 2021. "Endogenous stochastic arbitrage bubbles and the Black–Scholes model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 583(C).
    18. Rotundo, Giulia, 2014. "Black–Scholes–Schrödinger–Zipf–Mandelbrot model framework for improving a study of the coauthor core score," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 404(C), pages 296-301.
    19. Mauricio Contreras G. & Roberto Ortiz H, 2021. "Three little arbitrage theorems," Papers 2104.10187, arXiv.org.

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