Exponential time integration and Chebychev discretisation schemes for fast pricing of options

We consider exponential time differencing (ETD) schemes for the numerical pricing of options. Special treatments for the implementation of the boundary conditions that arise in finance are described. We show that only one explicit time step computation gives unconditional second order accuracy for European, Barrier and Butterfly spread options under both Black–Scholes geometric Brownian motion model and Merton's jump diffusion model with constant coefficients. In comparison, the commonly used Crank–Nicolson scheme is shown to be only conditionally stable due to lack of L0-stability. Finally, we describe how the use of spectral spatial discretisation based on a Chebychev grid point concentration strategy gives fourth order accurate option prices for both the Black–Scholes and Merton's jump–diffusion model.