LIE SYMMETRY APPROACH TO THE CEV MODEL

Abstract

Using a Lie algebraic approach we explicitly provide both the probability density function of the constant elasticity of variance (CEV) process and the fundamental solution for the associated pricing equation. In particular we reduce the CEV stochastic differential equation (SDE) to the SDE characterizing the Cox, Ingersoll and Ross (CIR) model, being the latter easier to treat. The fundamental solution for the CEV pricing equation is then obtained following two methods. We first recover a fundamental solution via the {\it invariant solution method}, while in the second approach we exploit Lie classical result on classification of linear partial differential equations (PDEs). In particular we find a map which transforms the pricing equation for the CIR model into an equation of the form $v_{\tau} = v_{yy} - \frac{A}{y^2}v$ whose fundamental solution is known. Then, by inversion, we obtain a fundamental solution for the CEV pricing equation.