  An interacting Black-Scholes model for option pricing, 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, was developed in [1]. 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 Schrodinger equation with hamiltonian operator H=H0+U(t), where H0 is
the equilibrium Black-Scholes hamiltonian and U(t) is the interaction term. If
the U(t) term is small enough, the interaction potential can be thought as a
perturbation, so one can compute the solution of the interacting Black-Scholes
equation in an approximate form by perturbation theory. In [2] by applying the
semi-classical considerations, an approximate solution of the non equilibrium
Black-Scholes equation for an arbitrary bubble shape f(t) was developed. Using
this semi-classical solution and the knowledge about the mispricing of the
financial data, one can determinate an equation, which solutions permit obtain
the 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 for calibrating and simulating option
pricing solutions in the presence of market imperfections, could help to
improve option pricing estimations.
