LIE SYMMETRY APPROACH TO THE CEV MODEL

Abstract: Using a Lie algebraic approach we explicitly provide both the probabilitydensity function of the constant elasticity of variance (CEV) process andthe fundamental solution for the associated pricing equation. In particular wereduce the CEV stochastic differential equation (SDE) to the SDE characterizingthe Cox, Ingersoll and Ross (CIR) model, being the latter easier to treat.The fundamental solution for the CEV pricing equation is then obtained followingtwo methods. We first recover a fundamental solution via the invariantsolution method, while in the second approach we exploit Lie classical result onclassification of linear partial differential equations (PDEs). In particular wefind a map which transforms the pricing equation for the CIR model into anequation of the form vτ = vyy− Ay2 v whose fundamental solution is known. Then,by inversion, we obtain a fundamental solution for the CEV pricing equation.