Abstract

An non-equilibrium Black-Scholes model, where the usual constant interest rate r is replaced by a stochastic time dependent rate r(t) of the form r(t) = r + f(t) dW/dt , accounting for market imperfections and prices non-alignment, is developed. The white noise amplitude f(t), called arbitrage bubble, generates a time dependent potential U(t) which changes the usual equilibrium dynamics of the traditional Black-Scholes model. The purpose of this article is to tackle the inverse problem, that is, is it possible to extract the time dependent potential U(t) and its associated bubble shape f(t) from the real empirical financial data? In order to give an answer to this question, the interacting Black-Scholes equation must be interpreted as a quantum Schrödinger equation with Hamiltonian operator H = H0 + U(t), where H0 is the equilibrium Black-Scholes Hamiltonian and U(t) is the interaction term. By using semi-classical considerations and the knowledge about the mispricing of the financial data, one can determinate an approximate functional form of the potential term U(t) and its associated bubble f(t). In all the studied cases, the non-equilibrium model performs a better estimation of the real data than the usual equilibrium model. It is expected that this new and simple methodology could help to improve option pricing estimations.