On the symmetries in the generalized Black-Scholes model with variable coefficients

Abstract or Introduction

According to previous research, the issue on the applicability of the original Black-Scholes model to the inverse quantity of price can be formulated as the argument of the symmetry between price and its inverse, whether there exists the set of real numbers as the drift and the volatility about the inverse quantity satisfying a certain system of stochastic differential equations. As the result of solving the equations in terms of such real numbers, it is revealed that there exist symmetries between not only them but also the coefficients of two equations. The aim of this article is to reveal in which cases these symmetries exist in the generalized Black-Scholes model, where the coeficients are deterministic or stochastic processes.