| Abstract: |
| The invariant approach is employed to solve the Cauchy value problem for the bond-pricing partial differential equation (PDE) of mathematics of finance. We first briefly review the invariant criteria for scalar second-order parabolic partial differential equation in two independent variables. The criteria is then utilized to reduce the bond-pricing equation to different Lie canonical forms. We find that the invariant approach aids in transforming
the bond-pricing equation to the second Lie canonical form and with a proper parametric selection, the bond-pricing PDE can be converted to the first Lie canonical form which is the classical heat equation. Different cases have been deduced for which the original equation can be reduced to the first and the second Lie canonical forms. For each of the cases, we also work out the transformations which map the bond-pricing equation into the heat equation and also to the second Lie canonical form. We construct the fundamental solutions for the
bond-pricing model via these transformations by utilizing the 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 bond-pricing
model with proper choice of terminal boundary conditions are also obtained. |
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