Boundary value methods with the Crank–Nicolson preconditioner for pricing options in the jump-diffusion model

Abstract

Under a jump-diffusion process, the option pricing function satisfies a partial integro-differential equation. A fourth-order compact scheme is used to discretize the spatial variable of this equation. The boundary value method is then utilized for temporal integration because of its unconditional stability and high-order accuracy. Two approaches, the local mesh refinement and the start-up procedure with refined step size, are raised to avoid the numerical malfunction brought by the nonsmooth payoff function. The GMRES method with a preconditioner which comes from the Crank–Nicolson formula is employed to solve the resulting large-scale linear system. Numerical experiments demonstrate the efficiency of the proposed method when pricing European and double barrier call options in the jump-diffusion model.

Keywords:

• jump-diffusion
• boundary value method
• preconditioner
• Crank–Nicolson time-marching scheme
• fourth-order compact scheme
• Toeplitz matrix

2000 AMS Subject Classifications :

• 65M06
• 65L05
• 65F10
• 91B28

Acknowledgements

This work was partially supported by the research grant 033/2009/A from FDCT of Macao and UL020/08-Y3/MAT/JXQ01/FST and RG057/09-10S/SHW/FST from the University of Macau. The authors are also grateful to the anonymous referees for their constructive comments and suggestions which substantially improved the content of this paper.