The relative efficiency of numerical methods for pricing American options under Lévy processes

ABSTRACT We analyze properties of prices of American options under Lévy processes and the related difficulties for design of accurate and efficient numerical methods for pricing of American options. The case of Lévy processes with an insignificant diffusion component and jump part of infinite activity but finite variation (the case most relevant in practice according to the empirical study in Carr et al (2002)) appears to be the most difficult. Several numerical methods suggested for this case are discussed and compared. It is shown that approximations by diffusions with embedded jumps may be too inaccurate unless the time to expiry is large. However, the fitting by a diffusion with embedded exponentially distributed jumps and a new finite difference scheme suggested in the paper can be used as good complements, which ensure accurate and fast calculation of the option prices both close to expiry and far from it.We demonstrate that if the time to expiry is two months or more, and the relative error 2–3% is admissible, then the fitting by a diffusion with embedded exponentially distributed jumps and the calculation of prices using the semi-explicit pricing procedure in Levendorskiˇõ (2004a) is the best choice.