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Superconvergence of the finite element solutions of the Black–Scholes equation
Authors:A Golbabai  LV Ballestra  D Ahmadian
Institution:1. School of Mathematics, Iran University of Science and Technology, Narmak, 16844 Tehran, Iran;2. Dipartimento di Economia, Seconda Università di Napoli, Corso Gran Priorato di Malta, 81043 Capua, Italy
Abstract: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.
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