Convergence analysis of Crank–Nicolson and Rannacher time-marching

This paper presents a convergence analysis of Crank–Nicolson and Rannacher time-marching methods which are often used in finite difference discretizations of the Black–Scholes equations. Particular attention is paid to the important role of Rannacher’s startup procedure, in which one or more initial timesteps use backward Euler timestepping, to achieve second-order convergence for approximations of the first and second derivatives. Numerical results confirm the sharpness of the error analysis which is based on asymptotic analysis of the behavior of the Fourier transform. The relevance to Black–Scholes applications is discussed in detail, with numerical results supporting recommendations on how to maximize the accuracy for a given computational cost.