A Lie symmetry analysis of the Black-scholes Merton finance model through modified local one-parameter transformations

Abstract

The thesis presents a new method of Symmetry Analysis of the Black-Scholes Merton Finance Model through modi ed Local one-parameter transformations. We determine the symmetries of both the one-dimensional and two-dimensional Black-Scholes equations through a method that involves the limit of in nitesimal ! as it approaches zero. The method is dealt with extensively in [23]. We further determine an invariant solution using one of the symmetries in each case. We determine the transformation of the Black-Scholes equation to heat equation through Lie equivalence transformations. Further applications where the method is successfully applied include working out symmetries of both a Gaussian type partial di erential equation and that of a di erential equation model of epidemiology of HIV and AIDS. We use the new method to determine the symmetries and calculate invariant solutions for operators providing them.