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

TL;DR

This paper demonstrates that applying a square root transformation to the time variable in the Crank-Nicolson scheme ensures convergence when solving the heat equation with delta initial conditions, improving accuracy in option pricing.

Contribution

The study introduces a time change transformation that guarantees convergence of the Crank-Nicolson scheme and achieves quadratic convergence under certain conditions, enhancing numerical stability.

Findings

The transformed scheme always converges, unlike the original.

Quadratic convergence is achieved below a critical ratio of time to space step.

The method improves option pricing calculations without Rannacher start-up steps.

Abstract

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.