The impact of a natural time change on the convergence of the Crank-Nicolson scheme

We first analyse the effect of a square root transformation to the time variable on the convergence of the Crank-Nicolson scheme when applied to the solution of the heat equation with Dirac delta function initial conditions. In the original variables, the scheme is known to diverge as the time step is reduced with the ratio of the time step to space step held constant and the value of this ratio controls how fast the divergence occurs. After introducing the square root of time variable we prove that the numerical scheme for the transformed partial differential equation now always converges and that the ratio of the time step to space step controls the order of convergence, quadratic convergence being achieved for this ratio below a critical value. Numerical results indicate that the time change used with an appropriate value of this ratio also results in quadratic convergence for the calculation of the price, delta and gamma for standard European and American options without the need for Rannacher start-up steps.