Convergence of Crank-Nicolson Scheme Combined with Wavelet Discretization for the Black-Scholes Equation

This paper is concerned with numerical solution of the Black-Scholes equation. We apply the Crank-Nicolson scheme for time discretization and Hermite cubic spline wavelets with four vanishing moments for space discretization. The advantages of this approach are higher order accuracy, a small number of iterations needed to resolve the problem with desired accuracy and a high potential in adaptive methods due to the four vanishing wavelet moments. Due to the data irregularities in the model, Crank-Nicolson scheme is often used together with Rannacher startup procedure to achieve second order convergence for approximations of the first derivatives. We numerically show that optimal convergence rate for the proposed scheme is obtained without using startup procedure. Moreover, it is necessary to apply only small number of GMRES iterations to achieve this results.