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, lambda, of the time step to space step held constant, and the value of lambda 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 lambda controls the order of convergence, quadratic convergence being achieved for lambda below a critical value. Numerical results indicate that the time change used with an appropriate value of lambda 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.