Solving financial differential equations using differentiation matrices

This paper illustrates the use of the differentiation matrix technique for solving differential equations in finance. The technique provides a compact and unified formulation for a variety of discretisation and time-stepping algorithms for solving problems in one and two dimensions. Using differentiation matrix models, we compare time-stepping algorithms for option pricing computations and present numerical results that show the advantage of the L-stable Alexander method over the Crank-Nicolson method. We also compare the efficiency of the spectral collocation and finite difference methods, and give numerical results that show spectral methods to be competitive for problems with smooth terminal conditions.