Classical and Quantum-Like Randomness and the Financial Market

  • Andrei Khrennikov
Part of the New Economic Windows book series (NEW)


The financial market is a complex dynamical system and, since the publication of the thesis of [1], there were performed numerous studies devoted to various aspects of random description of financial processes [2]. At the first stage of investigations Brownian motion was used to describe randomness of the financial market. This model provided a rather good approximation of some financial processes. However, later it became evident that the diversity of financial stochastic processes could not be reduced to Brownian motion. The next step was consideration of functionals of Brownian motion, especially, geometric Brownian motion [2]. Later there were considered other types of stochastic processes [2], in particular, general Levy processes.


Option Price Quantum Randomness Quantum Formalism Bohmian Mechanic Classical Probabilistic Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bachelier L. (1890) Ann. Sc. lcole Normale Superiere 111–17: 21–121.Google Scholar
  2. 2.
    Bell J.S. (1987) Speakable and unspeakable in quantum mechanics. Cambridge Univ. Press, Cambridge.Google Scholar
  3. 3.
    Choustova O. (2001) Pilot wave quantum model for the stock market. Scholar
  4. 4.
    Choustova O. (2004) Bohmian mechanics for financial processes. J. Modern Optics 51: 1111.ADSMathSciNetGoogle Scholar
  5. 5.
    Choustova O. (2007) Quantum Bohmian model for financial market. Physica A 374: 304–314.CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    d’Espagnat B. (1995) Veiled Reality. An analysis of present-day quantum mechanical concepts. Addison-Wesley.Google Scholar
  7. 7.
    Fama E.F. (1970) Journal of Finance 25, 383.CrossRefGoogle Scholar
  8. 8.
    Haven E. (2002) A discussion on embedding the Black-Scholes option pricing model in a quantum physics setting. Physica A 304: 507–524.MATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Haven E. (2003) A Black-Sholes Schrödinger option price: ‘bit’ versus ‘qubit’, Physica A 324: 201–206.MATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Haven E. (2003) An ‘h-Brownian motion’ of stochastic option prices, Physica A 344: 151–155.Google Scholar
  11. 11.
    Haven E. (2006) Bohmian mechanics in a macroscopic quantum system. In: Adenier G, Khrennikov AYu, and Nieuwenhuizen T. (eds) Foundations of Probability and Physics-3 810: 330–340. American Institute of Physics, Melville, New York.Google Scholar
  12. 12.
    Khrennikov A. Yu (1999) Interpretations of Probability. VSP Int. Sc. Publishers, Utrecht/Tokyo (second edition-2004).Google Scholar
  13. 13.
    Khrennikov A. Yu (2004) Information dynamics in cognitive, psychological, social, and anomalous phenomena. Kluwer Academic, Dordrecht.MATHGoogle Scholar
  14. 14.
    Kolmogoroff A.N. (1933) Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer Verlag, Berlin; reprinted: Kolmogorov A.N. (1956) Foundations of the Probability Theory. Chelsea Publ. Comp., New York.Google Scholar
  15. 15.
    Mandelbrot B., Hudson R. (2004) The (mis)behavior of markets. Basic Books Publication.Google Scholar
  16. 16.
    von Neumann J. (1955) Mathematical foundations of quantum mechanics. Princeton Univ. Press, Princeton, N.J.MATHGoogle Scholar
  17. 17.
    Piotrowski E.W., Sladkowski J. (2001) Quantum-like approach to financial risk: quantum anthropic principle. Scholar
  18. 18.
    Samuelson P.A. (1965) Industrial Management Review 6, 41.Google Scholar
  19. 19.
    Segal W. and Segal I.E. (1998) Proc. Nat. Acad. Sc. USA 95: 4072.MATHCrossRefADSGoogle Scholar
  20. 20.
    Segal, W. and Segal, I.E. (1998). The BlackScholes pricing formula in the quantum context. Proceedings of the National Academy of Sciences USA 95, 4072–4080.MATHCrossRefADSGoogle Scholar
  21. 21.
    Shimony A. (1993) Search for a naturalistic world view. Cambridge Univ. Press, Cambridge.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia 2009

Authors and Affiliations

  • Andrei Khrennikov
    • 1
  1. 1.International Center for Mathematical Modeling in Physics, Engineering and Cognitive ScienceVäxjö UniversitySweden

Personalised recommendations