Error Estimates for Backward Euler Finite Element Approximations of American Call Option Valuation

Abstract

We investigate the degenerate parabolic variational inequality arising from the local volatility model for the American stock call option with continuous dividend payout. The problem, originally defined on the positive semiaxis, is formulated in a bounded domain with exact Dirichlet boundary condition under natural assumptions about the regularity of the data. The smoothness of the exact solution of the formulated variational inequality, which is necessary for studying the accuracy of the backward Euler finite element approximation, is established. For an approximate solution, the error estimate $$O(h+\tau^{\min\{\alpha,3/4\}})$$ in the energy norm is obtained under the assumption that the coefficients of the differential operator are piecewise $$\alpha$$ -Holder-continuous with respect to time variable, where $$h$$ and $$\tau$$ denote the mesh parameters in space and time, respectively.