Superconvergence of the finite element solutions of the Black-Scholes equation

We investigate the performances of the finite element method in solving the Black-Scholes option pricing model. Such an analysis highlights that, if the finite element method is carried out properly, then the solutions obtained are superconvergent at the boundaries of the finite elements. In particular, this is shown to happen for quadratic and cubic finite elements, and for the pricing of European vanilla and barrier options. To the best of our knowledge, lattice-based approximations of the Black-Scholes model that exhibit nodal superconvergence have never been observed so far, and are somehow unexpected, as the solutions of the associated partial differential problems have various kinds of irregularities.

► We study the performances of the finite element method (FEM) applied to the Black-Scholes model. ► We show how to obtain nodal superconvergence. ► Nodal superconvergence is achieved even if the solutions have various kinds of irregularities. ► At the FEM boundary nodes the errors are extremely small (even with FEM basis of relatively low degree). ► Both vanilla and barrier options are considered.