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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.
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.