IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i21p2777-d670455.html
   My bibliography  Save this article

The Probability Flow in the Stock Market and Spontaneous Symmetry Breaking in Quantum Finance

Author

Listed:
  • Ivan Arraut

    (Lee Shau Kee School of Business and Administration, The Open University of Hong Kong, 30 Good Shepherd Street, Homantin, Kowloon, Hong Kong, China
    These authors contributed equally to this work.)

  • João Alexandre Lobo Marques

    (FBL, University of Saint Joseph Estrada Marginal da Ilha Verde, 14-17, Macao, China
    These authors contributed equally to this work.)

  • Sergio Gomes

    (FBL, University of Saint Joseph Estrada Marginal da Ilha Verde, 14-17, Macao, China
    These authors contributed equally to this work.)

Abstract

The spontaneous symmetry breaking phenomena applied to Quantum Finance considers that the martingale state in the stock market corresponds to a ground (vacuum) state if we express the financial equations in the Hamiltonian form. The original analysis for this phenomena completely ignores the kinetic terms in the neighborhood of the minimal of the potential terms. This is correct in most of the cases. However, when we deal with the martingale condition, it comes out that the kinetic terms can also behave as potential terms and then reproduce a shift on the effective location of the vacuum (martingale). In this paper, we analyze the effective symmetry breaking patterns and the connected vacuum degeneracy for these special circumstances. Within the same scenario, we analyze the connection between the flow of information and the multiplicity of martingale states, providing in this way powerful tools for analyzing the dynamic of the stock markets.

Suggested Citation

  • Ivan Arraut & João Alexandre Lobo Marques & Sergio Gomes, 2021. "The Probability Flow in the Stock Market and Spontaneous Symmetry Breaking in Quantum Finance," Mathematics, MDPI, vol. 9(21), pages 1-18, November.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:21:p:2777-:d:670455
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/21/2777/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/21/2777/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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..
    2. Hélyette Geman & Marc Yor, 1993. "Bessel Processes, Asian Options, And Perpetuities," Mathematical Finance, Wiley Blackwell, vol. 3(4), pages 349-375, October.
    3. Johnson, Herb & Shanno, David, 1987. "Option Pricing when the Variance Is Changing," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(2), pages 143-151, June.
    4. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    5. Jones, E. Philip, 1984. "Option arbitrage and strategy with large price changes," Journal of Financial Economics, Elsevier, vol. 13(1), pages 91-113, March.
    6. Scott, Louis O., 1987. "Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(4), pages 419-438, December.
    7. Ivan Arraut & Alan Au & Alan Ching-biu Tse & Carlos Segovia, 2019. "The connection between multiple prices of an Option at a given time with single prices defined at different times: The concept of weak-value in quantum finance," Papers 1905.05813, arXiv.org.
    8. Poterba, James M & Summers, Lawrence H, 1986. "The Persistence of Volatility and Stock Market Fluctuations," American Economic Review, American Economic Association, vol. 76(5), pages 1142-1151, December.
    9. Arraut, Ivan & Au, Alan & Tse, Alan Ching-biu & Segovia, Carlos, 2019. "The connection between multiple prices of an Option at a given time with single prices defined at different times: The concept of weak-value in quantum finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 526(C).
    10. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    11. Lamoureux, Christopher G & Lastrapes, William D, 1993. "Forecasting Stock-Return Variance: Toward an Understanding of Stochastic Implied Volatilities," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 293-326.
    12. 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.
    13. Baaquie, Belal E. & Corianò, Claudio & Srikant, Marakani, 2004. "Hamiltonian and potentials in derivative pricing models: exact results and lattice simulations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 334(3), pages 531-557.
    14. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    15. Ivan Arraut & Alan Au & Alan Ching-biu Tse, 2020. "Spontaneous symmetry breaking in Quantum Finance," Papers 2011.05278, arXiv.org.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ghysels, E. & Harvey, A. & Renault, E., 1995. "Stochastic Volatility," Papers 95.400, Toulouse - GREMAQ.
    2. David S. Bates, 1995. "Testing Option Pricing Models," NBER Working Papers 5129, National Bureau of Economic Research, Inc.
    3. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    4. Carl Chiarella & Xue-Zhong He & Christina Sklibosios Nikitopoulos, 2015. "Derivative Security Pricing," Dynamic Modeling and Econometrics in Economics and Finance, Springer, edition 127, number 978-3-662-45906-5, July-Dece.
    5. Ivan Arraut & Alan Au & Alan Ching-biu Tse, 2020. "On the multiplicity of the martingale condition: Spontaneous symmetry breaking in Quantum Finance," Papers 2004.11270, arXiv.org.
    6. Mondher Bellalah, 2009. "Derivatives, Risk Management & Value," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 7175, January.
    7. Jun Ma, 2009. "A Stochastic Correlation Model with Mean Reversion for Pricing Multi-Asset Options," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 16(2), pages 97-109, June.
    8. Lars Stentoft, 2008. "Option Pricing using Realized Volatility," CREATES Research Papers 2008-13, Department of Economics and Business Economics, Aarhus University.
    9. Robert Azencott & Yutheeka Gadhyan & Roland Glowinski, 2014. "Option Pricing Accuracy for Estimated Heston Models," Papers 1404.4014, arXiv.org, revised Jul 2015.
    10. Rombouts, Jeroen V.K. & Stentoft, Lars, 2015. "Option pricing with asymmetric heteroskedastic normal mixture models," International Journal of Forecasting, Elsevier, vol. 31(3), pages 635-650.
    11. Gonçalo Faria & João Correia-da-Silva, 2014. "A closed-form solution for options with ambiguity about stochastic volatility," Review of Derivatives Research, Springer, vol. 17(2), pages 125-159, July.
    12. David Heath & Simon Hurst & Eckhard Platen, 1999. "Modelling the Stochastic Dynamics of Volatility for Equity Indices," Research Paper Series 7, Quantitative Finance Research Centre, University of Technology, Sydney.
    13. Ivan Arraut & Joao Alexandre Lobo Marques & Sergio Gomes, 2022. "The probability flow in the Stock market and Spontaneous symmetry breaking in Quantum Finance," Papers 2206.07130, arXiv.org.
    14. Mikhail Chernov & Eric Ghysels, 1998. "What Data Should Be Used to Price Options?," CIRANO Working Papers 98s-22, CIRANO.
    15. Huang, Yu Chuan & Chen, Shing Chun, 2002. "Warrants pricing: Stochastic volatility vs. Black-Scholes," Pacific-Basin Finance Journal, Elsevier, vol. 10(4), pages 393-409, September.
    16. Peter A. Abken & Saikat Nandi, 1996. "Options and volatility," Economic Review, Federal Reserve Bank of Atlanta, vol. 81(Dec), pages 21-35.
    17. Kim, In Joon & Kim, Sol, 2004. "Empirical comparison of alternative stochastic volatility option pricing models: Evidence from Korean KOSPI 200 index options market," Pacific-Basin Finance Journal, Elsevier, vol. 12(2), pages 117-142, April.
    18. Yacine Ait-Sahalia & Robert Kimmel, 2004. "Maximum Likelihood Estimation of Stochastic Volatility Models," NBER Working Papers 10579, National Bureau of Economic Research, Inc.
    19. Peter Christoffersen & Kris Jacobs, 2004. "Which GARCH Model for Option Valuation?," Management Science, INFORMS, vol. 50(9), pages 1204-1221, September.
    20. Grunbichler, Andreas & Longstaff, Francis A., 1996. "Valuing futures and options on volatility," Journal of Banking & Finance, Elsevier, vol. 20(6), pages 985-1001, July.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2021:i:21:p:2777-:d:670455. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.