Accurate and efficient pricing of vanilla stock options via the Crandall–Douglas scheme

Abstract

In this paper, we apply the Crandall–Douglas scheme [Quarterly of Applied Mathematics 13 (1955) 318–320, Journal of Mathematics and Physics 35 (1956) 145–151] for the heat equation to the computation of vanilla stock option prices. This method has the advantage of being second-order accurate in time and fourth-order accurate in “space” (i.e., asset price) and can even be made sixth-order accurate if we restrict the size of the time step. Although there is an exact solution for European options, we study their numerical approximation so as to determine suitable discretization parameters and permit reasonable placement of the far-field boundary. We then turn to American options where there are no analytical pricing formulae. Discretization of the associated free boundary problem, when expressed as a variational inequality, leads to a linear complementarity problem at each time step. These subproblems are solved in linear time by the Elliott–Ockendon algorithm [Weak and Variational Methods for Moving Boundary Problems, Pitman, 1982, pp. 115–117]. The net result is an option pricing methodology which is both very accurate and highly efficient. We present the application of our numerical scheme to calls paying continuous dividends as well as to puts, in both cases permitting early exercise of the option.