Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
See the monographs [9, 12], for an extended bibliography. We especially emphasize the interest for modeling the behavior of financial agents. See the aforementioned monographs and papers of Piotrowski and Sladkowski [15, 16], Choustova [6, 7], Accardi et al. [1], Asano et al. [2, 3] and authors’ paper [13]. We also want to call the reader’s attention to one of the pioneering articles which justifies the usage of methods of quantum mechanics in social science [11].
We can define put options in a similar way. We do not need that definition in this paper.
Such expansion carries the name of ‘Itô Lemma’. The expansion is peculiar since it occurs within a stochastic context.
Definition 2 below explains the notion of wave number in an economics context.
We want to thank one of the referees for pointing this out.
A barrier symbolizes a level of potential energy. See also Fig. 1 below.
Position could be the price of the asset, like a stock.
Mass is incorporated into the volatility parameter we mention in Definition 2 below. Please see also the discussion, immediately under Eq. (15), on this issue.
We thank one of the referees of this paper for bringing up the important point of quantum discreteness.
The free particle, V=0 puts in peril spatial localization and this condition is intimately related to the existence of so called ‘hermiticity’. It can be interesting to connect the existence of arbitrage with the existence of hermiticity. We do not expand on this in this paper.
For instance, a very specific investment house may have spotted an arbitrage opportunity.
Time scales for arbitrage opportunities can be very short.
Again, within a classical mechanical context, such transmission is fully impossible.
We thank one of the referees for raising this important issue.
But please see the next section (Sect. 8) in this paper for an argument why we need to be very careful in using the Fourier (integration/transformation) apparatus in the tunnelling environment.
Remark that the transmission coefficient only exists when κ is non-zero. But κ is non-zero only when there is arbitrage, since V>E only can make sense when V is not zero.
References
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)
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)
Asano, M., Basieva, I., Khrennikov, A., Ohya, M., Tanaka, Y.: Quantum like dynamics of decision making. Physica A 391, 2083–2099 (2012)
Bailey, R.: The Economics of Financial Markets. Cambridge University Press, Cambridge (2005)
Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973)
Choustova, O.A.: Bohmian mechanics for financial processes. J. Mod. Opt. 51, 1111 (2004)
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)
Harrison, J.M., Kreps, D.M.: Martingales and arbitrage in multiperiod securities markets. J. Econ. Theory 20, 381–408 (1979)
Haven, E., Khrennikov, A.: Quantum Social Science. Cambridge University Press, Cambridge (2013)
Hull, J.: Options, Futures and Other Derivatives. Prentice Hall, New York (2006)
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)
Khrennikov, A. Yu.: Ubiquitous Quantum Structure: From Psychology to Finance. Springer, Berlin (2010)
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)
Paul, W., Baschnagel, J.: Stochastic Processes: From Physics to Finance. Springer, Berlin (2000)
Piotrowski, E.W., Sladkowski, J.: An invitation to quantum game theory. Int. J. Theor. Phys. 42, 1089–1099 (2003)
Piotrowski, E.W., Sladkowski, J.: Trading by quantum rules: quantum anthropic principle. Int. J. Theor. Phys. 42, 1101–1106 (2003)
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)
Plotnitsky, A.: Reading Bohr: Physics and Philosophy. Springer, Dordrecht (2006)
Plotnitsky, A.: Epistemology and Probability: Bohr, Heisenberg, Schrödinger, and the Nature of Quantum-Theoretical Thinking. Springer, Berlin (2009)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Haven, E., Khrennikov, A. Quantum-Like Tunnelling and Levels of Arbitrage. Int J Theor Phys 52, 4083–4099 (2013). https://doi.org/10.1007/s10773-013-1722-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10773-013-1722-0