Symmetry-based solutions of bond pricing classical vasicek and cox-ingersoll-ross models from financial mathematics

The bond pricing models of Vasicek and Cox-Ingersoll-Ross (CIR) are solved using the invariant approach. The invariance criteria is employed to the linear (1+1) parabolic partial differential equations (PDEs), namely, Vasicek and CIR models in order to perform reduction into one of the four Lie canonical forms. The invariant approach helps in transforming the PDE representing the Vasicek model into first Lie canonical form which is the 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 PDE can be converted to the first Lie canonical form. For both the Vasicek and CIR models, we obtain the transformations which map these PDEs into the heat equation and also to the second Lie canonical form. We construct the 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 conditions are also deduced.