Numerical pricing of American options under infinite activity Lévy processes

Under infinite activity Lévy models, American option prices can be obtained by solving a partial integro‐differential equation (PIDE), which has a singular kernel. With increasing degree of singularity, standard time‐stepping techniques may encounter difficulties. This study examines exponential time integration (ETI) for solving this problem and the performance of this scheme is compared with the Crank–Nicolson (CN) method and an implicit–explicit method in conjunction with an extrapolation (IMEX‐Extrap), in terms of computational speed and convergence orders. These findings indicate that ETI is faster and more accurate among PIDE‐based methods for solving the system of ordinary differential equations resulting from spatial discretization of the PIDE. For very singular problems, it is shown that the IMEX‐Extrap scheme becomes unfavorable compared with the other schemes as it is relatively more time consuming and the global convergence deteriorates from quadratic to linear, whereas the ETI scheme yields both point‐wise and global quadratic convergence. For illustration, under the infinite variation process, the IMEX‐Extrap achieves a precision of the order of 10-super-−4 in 663.016 s, whereas for the same set of parameters, the CN method and the ETI scheme reach an accuracy of the order of 10-super-−5 in 237.891 s and 22.772 s, respectively. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:809–829, 2011