Diffusion approximation of Lévy processes with a view towards finance

Handle URI:
http://hdl.handle.net/10754/561696
Title:
Diffusion approximation of Lévy processes with a view towards finance
Authors:
Kiessling, Jonas; Tempone, Raul ( 0000-0003-1967-4446 )
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.
KAUST Department:
Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Stochastic Numerics Research Group
Publisher:
De Gruyter
Journal:
Monte Carlo Methods and Applications
Issue Date:
Jan-2011
DOI:
10.1515/MCMA.2011.003
Type:
Article
ISSN:
09299629
Appears in Collections:
Articles; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorKiessling, Jonasen
dc.contributor.authorTempone, Raulen
dc.date.accessioned2015-08-03T09:02:30Zen
dc.date.available2015-08-03T09:02:30Zen
dc.date.issued2011-01en
dc.identifier.issn09299629en
dc.identifier.doi10.1515/MCMA.2011.003en
dc.identifier.urihttp://hdl.handle.net/10754/561696en
dc.description.abstractLet 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.en
dc.publisherDe Gruyteren
dc.subjectA posteriori error estimatesen
dc.subjectAdaptivityen
dc.subjectDiffusion approximationen
dc.subjectError controlen
dc.subjectError expansionen
dc.subjectInfinite activityen
dc.subjectLévy processen
dc.subjectMathematical financeen
dc.subjectMonte carloen
dc.subjectWeak approximationen
dc.titleDiffusion approximation of Lévy processes with a view towards financeen
dc.typeArticleen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentStochastic Numerics Research Groupen
dc.identifier.journalMonte Carlo Methods and Applicationsen
dc.contributor.institutionInstitute for Mathematics, Royal Institute of Technology, S-10044 Stockholm, Swedenen
kaust.authorTempone, Raulen
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