International Journal of Theoretical Physics

, Volume 52, Issue 11, pp 4083–4099 | Cite as

Quantum-Like Tunnelling and Levels of Arbitrage

  • Emmanuel Haven
  • Andrei KhrennikovEmail author


We apply methods of wave mechanics to financial modelling. We proceed by assigning a financial interpretation to wave numbers. This paper makes a plea for the use of the concept of ‘tunnelling’ (in the mathematical formalism of quantum mechanics) in the modelling of financial arbitrage. Financial arbitrage is a delicate concept to model in social science (i.e. in this case economics and finance) as its presence affects the precision of benchmark financial asset prices. In this paper, we attempt to show how ‘tunnelling’ can be used to positive effect in the modelling of arbitrage in a financial asset pricing context.


Quantum-like models Finance Wave numbers Arbitrage 


  1. 1.
    Accardi, L., Khrennikov, A., Ohya, M.: Quantum Markov model for data from Shafir–Tversky experiments in cognitive psychology. Open Syst. Inf. Dyn. 16, 371–385 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Asano, M., Ohya, M., Tanaka, Y., Basieva, I., Khrennikov, A.: Quantum-like model of brain’s functioning: decision making from decoherence. J. Theor. Biol. 281, 56–64 (2011) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Asano, M., Basieva, I., Khrennikov, A., Ohya, M., Tanaka, Y.: Quantum like dynamics of decision making. Physica A 391, 2083–2099 (2012) ADSCrossRefGoogle Scholar
  4. 4.
    Bailey, R.: The Economics of Financial Markets. Cambridge University Press, Cambridge (2005) CrossRefGoogle Scholar
  5. 5.
    Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973) CrossRefGoogle Scholar
  6. 6.
    Choustova, O.A.: Bohmian mechanics for financial processes. J. Mod. Opt. 51, 1111 (2004) MathSciNetADSGoogle Scholar
  7. 7.
    Choustova, O.: Application of Bohmian mechanics to dynamics of prices of shares: stochastic model of Bohm-Vigier from properties of price trajectories. Int. J. Theor. Phys. 47, 252–260 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Harrison, J.M., Kreps, D.M.: Martingales and arbitrage in multiperiod securities markets. J. Econ. Theory 20, 381–408 (1979) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Haven, E., Khrennikov, A.: Quantum Social Science. Cambridge University Press, Cambridge (2013) CrossRefGoogle Scholar
  10. 10.
    Hull, J.: Options, Futures and Other Derivatives. Prentice Hall, New York (2006) Google Scholar
  11. 11.
    Khrennikov, A. Yu.: Classical and quantum mechanics on information spaces with applications to cognitive, psychological, social and anomalous phenomena. Found. Phys. 29, 1065–1098 (1999) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Khrennikov, A. Yu.: Ubiquitous Quantum Structure: From Psychology to Finance. Springer, Berlin (2010) CrossRefGoogle Scholar
  13. 13.
    Khrennikov, A. Yu., Haven, E.: Quantum mechanics and violations of the sure-thing principle: the use of probability interference and other concepts. J. Math. Psychol. 53, 378–388 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Paul, W., Baschnagel, J.: Stochastic Processes: From Physics to Finance. Springer, Berlin (2000) Google Scholar
  15. 15.
    Piotrowski, E.W., Sladkowski, J.: An invitation to quantum game theory. Int. J. Theor. Phys. 42, 1089–1099 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Piotrowski, E.W., Sladkowski, J.: Trading by quantum rules: quantum anthropic principle. Int. J. Theor. Phys. 42, 1101–1106 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Plotnitsky, A.: “This is an extremely funny thing, something must be hidden behind that”: quantum waves and quantum probability with Erwin Schrödinger. In: Foundations of Probability and Physics-3. Ser. Conference Proceedings, vol. 750, pp. 388–408. American Institute of Physics, Melville (2005) Google Scholar
  18. 18.
    Plotnitsky, A.: Reading Bohr: Physics and Philosophy. Springer, Dordrecht (2006) Google Scholar
  19. 19.
    Plotnitsky, A.: Epistemology and Probability: Bohr, Heisenberg, Schrödinger, and the Nature of Quantum-Theoretical Thinking. Springer, Berlin (2009) Google Scholar
  20. 20.
    Tan, A.: Long memory stochastic volatility and a risk minimization approach for derivative pricing an hedging. Ph.D. Thesis, School of Mathematics University of Manchester (UK) (2005) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Management and Institute of FinanceUniversity of LeicesterLeicesterUK
  2. 2.International Centre for Mathematical ModellingLinnaeus UniversityVäxjöSweden

Personalised recommendations