Diffusion approximation of Lévy processes with a view towards finance

Jonas Kiessling, Raul Tempone

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

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.
Original languageEnglish (US)
Pages (from-to)11-45
Number of pages35
JournalMonte Carlo Methods and Applications
Volume17
Issue number1
DOIs
StatePublished - Jan 2011

ASJC Scopus subject areas

  • Statistics and Probability
  • Applied Mathematics

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