Calibration and Simulation of Arbitrage Effects in a Non-Equilibrium Quantum Black-Scholes Model by Using Semi-Classical Methods

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 ) W ( t ), 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 Schrodinger equation with Hamiltonian operator H = H 0 + U ( t ), where H 0 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.