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    Diffusion approximation of Lévy processes with a view towards finance

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    Type
    Article
    Authors
    Kiessling, Jonas
    Tempone, Raul cc
    KAUST Department
    Applied Mathematics and Computational Science Program
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Stochastic Numerics Research Group
    Date
    2011-01
    Permanent link to this record
    http://hdl.handle.net/10754/561696
    
    Metadata
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    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
    Publisher
    Walter de Gruyter GmbH
    Journal
    Monte Carlo Methods and Applications
    DOI
    10.1515/MCMA.2011.003
    ae974a485f413a2113503eed53cd6c53
    10.1515/MCMA.2011.003
    Scopus Count
    Collections
    Articles; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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