Adaptive Finite Element Solution of 1D European Option Pricing Problems - Jackson, N; Suli, E (Unspecified, 1997)
We present a piecewise Hermite cubic adaptive finite element method for solving a generalised European Black-Scholes problem to guaranteed accuracy. Specifically, we prove a residual-based a posteriori error bound in the $L^{2}(Ω)$-norm, at contract issue, for a continuous Galerkin approximation to the solution using Galerkin orthogonality and weighted strong stability of an associated dual problem. We use this bound to construct an adaptive algorithm to generate a space-time discretisation which ensures that the error norm is less than a given tolerance. We demonstrate the speed and accuracy of our method through example pricings.