Diffusion approximation of Lévy processes with a view towards finance

Let the (log-)prices of a collection of securities be given by a d–dimensional Lévy process Xt having infinite activity and a smooth density. The value of a European contract with payoff g(x) maturing at T is determined by E[g(XT )]. Let X̄T be a finite activity approximation to XT , where diffusion is introduced to approximate jumps smaller than a given truncation level ! > 0. The main result of this work is a derivation of an error expansion for the resulting model error, E[g(XT )−g(X̄T )], with computable leading order term. Our estimate depends both on the choice of truncation level ! and the contract payoff g, and it is valid even when g is not continuous. Numerical experiments confirm that the error estimate is indeed a good approximation of the model error. Using similar techniques we indicate how to construct an adaptive truncation type approximation. Numerical experiments indicate that a substantial amount of work is to be gained from such adaptive approximation. Finally, we extend the previous model error estimates to the case of Barrier options, which have a particular path dependent structure.