Diffusion approximation of Lévy processes with a view towards finance
Type
ArticleAuthors
Kiessling, JonasTempone, Raul

KAUST Department
Applied Mathematics and Computational Science ProgramComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Stochastic Numerics Research Group
Date
2011-01Permanent link to this record
http://hdl.handle.net/10754/561696
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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.2011.003Publisher
Walter de Gruyter GmbHae974a485f413a2113503eed53cd6c53
10.1515/MCMA.2011.003