J. Kiessling and R. Tempone. Computable error estimates of a finite difference scheme for option pricing in exponential Levy models. BIT Numerical Mathematics, December 2014, Volume 54, Issue 4, pp 1023-1065
J. Kiessling And R. Tempone
Levy process, infinite activity, diffusion approximation, parabolic integro-differential equation, weak approximation, error expansion, a posteriori error estimates, finite difference method, option pricing, jump-diffusion models
Option prices in exponential L´evy models solve certain partial integrodifferential equations (PIDEs). This work focuses on a finite difference scheme that is suitable for solving such PIDEs. The scheme was introduced in [Cont and Voltchkova, SIAM J. Numer. Anal., 43(4):1596–1626, 2005]. The main results of this work are new estimates of the dominating error terms, namely the time and space discretization errors. In addition, the leading order terms of the error estimates are determined in computable form. The payoff is only assumed to satisfy an exponential growth condition, it is not assumed to be Lipschtitz continuous as in previous works. If the underlying Lévy process has infinite jump activity, then the jumps smaller than some ε> 0 are approximated by diffusion. The resulting diffusion approximation error is also estimated, with leading order term in computable form, as well as its effect on space and time discretization errors. Consequently, it is possible to determine how to jointly choose the space and time grid sizes and the parameter ε.
ISSN: DOI 10.1007/s10543-014-0490-4