Galerkin FEM for Black-Scholes PDE

The main method for numerical solutions to PDEs in finance is the Finite Difference method (FDM). We show how an alternative method, the Finite Element method (FEM) can be used instead. The main strength of FEM is arguably its flexibility given by the grid construction which is no longer a set of isolated points but a grid of functions. This compared to FDM, in particular, means that no interpolation is needed as the value of the contingent claim is given everywhere in the space domain by a local function. The introductory exposition is dedicated to a general ODE and then moves to a Galerkin FEM formulation applied to a Black-Scholes PDE.