This work presents some results on the Green Functions of the Damped Wave Equation. We exhibit its fundamental solution and we study it. This fundamental solution allows us to obtain the solution of the initial value problem when the initial conditions are general distributions. We also give conditions on these initial conditions that insure us uniqueness of solution. To this purpose we develop some theory about the semigroup, as an application, and we finally apply our results to the price evolution of an asset in a Financial Model we propose.