A Superconvergent Fitted Finite Volume Method for Black–Scholes Equations Governing European and American Option Valuation

Abstract

We develop a superconvergent fitted finite volume method for a degenerate nonlinear penalized Black–Scholes equation arising in the valuation of European and American options, based on the fitting idea in Wang [IMA J Numer Anal 24 (2004), 699–720]. Unlike conventional finite volume methods in which the dual mesh points are naively chosen to be the midpoints of the subintervals of the primal mesh, we construct the dual mesh judiciously using an error representation for the flux interpolation so that both the approximate flux and solution have the second-order accuracy at the mesh points without any increase in computational costs. As the equation is degenerate, we also show that it is essential to refine the meshes locally near the degenerate point in order to maintain the second-order accuracy. Numerical results for both European and American options with constant and nonconstant coefficients will be presented to demonstrate the superconvergence of the method.