A Fast Numerical Algorithm for pricing of American Call Options under Jump-Diffusions‎: ‎An Artificial Boundary Approach

‎In this paper‎, ‎we propose a new method to approximate the price of an American call option written on a dividend-paying risky underlying following a jump-diffusion process‎. ‎We extend the methodology proposed in [H‎. ‎Han and X‎. ‎Wu‎, ‎A fast numerical method for the Black-Scholes equation of American options‎, ‎SIAM J‎. ‎Numer‎. ‎Anal‎. ‎41 (6) (2003) 2081-2095] to the case of parabolic partial integro-differential equations (PIDEs)‎. ‎The integral term in the PIDE arising from this model can also be considered as a driving term‎, ‎then the integro-differential equation can be viewed as a parabolic PDE‎. ‎Using this approach‎, ‎we introduce an exact non-local boundary condition on a suitably defined artificial (transparent) boundary introduced to reduce the infinite ``physical domain into a finite ``computational one‎. ‎We then develop a Crank-Nicolson scheme to solve the PIDE along with the artificial‎ ‎boundary condition‎. ‎Our results show that the proposed approach is efficient and gives a better‎ ‎accuracy than other alternatives from the literature‎.