Invariant criteria for the zero-coupon bond pricing Vasicek and Cox-Ingersoll-Ross Models

Abstract

The zero coupon bond pricing Vasicek and Cox-Ingersoll-Ross (CIR) interest rate models are solved using the invariant approach. The invariance criteria is employed on the linear (1+1) parabolic partial differential equations corresponding to the Vasicek and CIR models in order to perform reduction into one of the four Lie canonical forms. The invariant approach helps in transforming the partial differential equation representing the Vasicek model into the first Lie canonical form which is the classical heat equation. We also find that the invariant method aids in transforming the CIR model into the second Lie canonical form and with a proper parametric selection, the CIR equation can be converted to the first Lie canonical form. For both the Vasicek and CIR models, we obtain the transformations which map these equations into the heat equation and also to the second Lie canonical form. We constructthe fundamental solutions for the Vasicek and CIR models via these transformations by utilizing the well-known fundamental solutions of the classical heat equation as well as solution to the second Lie canonical form. Finally, the closed-form analytical solutions of the Cauchy initial value problems of the Vasicek and CIR models with suitable choice of terminal boundary conditions are also deduced.