Smoothness conditions and symmetries of partial differential equations

Abstract

We obtain a solution of the Black-Scholes equation with a non-smooth bound- ary condition using symmetry methods. The Black-Scholes equation along with its boundary condition are rst transformed into the one dimensional heat equation and an initial condition respectively. We then nd an appro- priate general symmetry generator of the heat equation using symmetries of the heat equation and the fundamental solution of the heat equation. The method we use to nd the symmetry generator is such that the boundary condition is left invariant and yet the symmetry can still be used to solve the heat equation. We then use the help of Mathematica to nd the solution to the heat equation. Then the solution is then transformed backwards to a solution of the Black-Scholes equation using the same change of variables that were used for the forward transformations. The solution is then nally checked if it satis es the boundary condition of the Black-Scholes equation.