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

*Mauricio Contreras*,
*Rely Pellicer*,
*Marcelo Villena* and
*Aaron Ruiz*

*Physica A: Statistical Mechanics and its Applications*, 2010, vol. 389, issue 23, 5447-5459

**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.

**Keywords:** Black–Scholes model; Arbitrage; Option pricing; Quantum mechanics; Semi-classical methods (search for similar items in EconPapers)

**Date:** 2010

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**Persistent link:** https://EconPapers.repec.org/RePEc:eee:phsmap:v:389:y:2010:i:23:p:5447-5459

**DOI:** 10.1016/j.physa.2010.08.018

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