Diffusion approximation of Lévy processes with a view towards finance

Diffusion approximation of Lévy processes with a view towards finance

Kiessling, Jonas; Tempone, Raúl "Diffusion approximation of Lévy processes with a view towards finance."

Monte Carlo Methods Appl. 17 (2011), no. 1, 11–45.

Kiessling, Jonas; Tempone, Raúl
Lévy process; infinite activity; diffusion approximation; Monte Carlo; weak approximation; error expansion; a posteriori error estimates; adaptivity; error control; mathematical finance
2011
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  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, , 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.

ISSN (Print) 0929-9629