Numerical Analysis Group Research Report NA-97/5

Adaptive Finite Element Solution of 1D European Option Pricing Problems

Nicholas Jackson and Endre Süli

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 L²(Omega)-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.

Key words and phrases:

Black-Scholes, a posteriori error analysis, adaptive finite element method, guaranteed accuracy

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