Diffusion approximation of Lévy processes with a view towards finance

Kiessling, Jonas; Tempone, Raul

Diffusion approximation of Lévy processes with a view towards finance

Type

Article

Authors

Kiessling, Jonas
Tempone, Raul

KAUST Department

Applied Mathematics and Computational Science Program
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Stochastic Numerics Research Group

Date

2011-01

Abstract

Let the (log-)prices of a collection of securities be given by a d-dimensional Lévy process X t 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(X T)]. Let X̄ T be a finite activity approximation to X T, 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(X T) - g(X̄ T)], 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. © de Gruyter 2011.

Citation

Kiessling, J., & Tempone, R. (2011). Diffusion approximation of Lévy processes with a view towards finance. Monte Carlo Methods and Applications, 17(1). doi:10.1515/mcma.2011.003