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

Abstract

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)Ẇ (t), 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 Schrödinger 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. 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.

Three little arbitrage theorems

“…Note that in this case, the solution of the interacting Black-Scholes equation [equation (34)] differs from the solution of the free equation [equation (40)] at maturity. In fact, it is a series of all the free Greeks.…”

“…Third arbitrage theorem: Let f (τ ) be an arbitrage bubble that acts in the time interval 0 < τ < T, and let A N (τ ) be the arbitrage number of f at time τ . Then, the solution π (S, τ ) of the interacting Black-Scholes equation [equation (34)] is just the solution to a free Black-Scholes equation with the variable interest rate…”

“…Thus, arbitrage bubbles can be characterized by a finite timespan and a constant height that measures the deviation from the Black-Scholes model's prices. Note that in general, a time-dependent arbitrage bubble f = f (t) can be determined approximately from the empirical financial data [34] by using semi-classical methods [35].…”

Endogenous stochastic arbitrage bubbles and the Black–Scholes model

“…To obtain a solution with N S = 0 and N V = 0, the determinant associated to the matrix form of this system (19) must be equal to zero; that is,…”

“…is a potential term that is equivalent to an electromagnetic potential that is induced by the arbitrage bubble f (S, t). An approximate solution of this equation for an arbitrary bubble form f (S, t) is given in [18] and a method to determine the bubble f from the real financial data is proposed in [19]. The resonances that appear in the model are also discussed in [20].…”