The Cox-Ingersoll-Ross (CIR) model is a short-rate model and is widely used in the finance field to predict the movement of interest rates in bond pricing models. This paper exploited Lie symmetry analysis to solve the generalized CIR model by determining the infinitesimal generators. Lie symmetry is one of the powerful tools to solve the partial differential equation (PDE) analytically by reducing the PDE into a lower form. Besides, an optimal system of one-dimensional subalgebras is constructed and then used to reduce the generalized CIR equation by introducing the similarity variables. Lastly, the invariant solutions are obtained by solving the reduced equation.

In this paper, considering the variational inequality model for American put option with dividends under Black-Scholes model, solving the variational equation by the finite difference method and the splitting method in time, numerical experiments have verified the effectiveness of the algorithm.