Diffusion approximation of Lévy processes with a view towards finance

Kiessling Jonas () and Tempone Raúl ()
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Kiessling Jonas: Institute for Mathematics, Royal Institute of Technology, S-10044 Stockholm, Sweden.
Tempone Raúl: Applied Mathematics and Computational Sciences, 4700 King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia.

Monte Carlo Methods and Applications, 2011, vol. 17, issue 1, 11-45

Abstract: Let the (log-)prices of a collection of securities be given by a d-dimensional Lévy process Xt 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(XT)]. Let be a finite activity approximation to XT, 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, , 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.

Keywords: Lévy process; infinite activity; diffusion approximation; Monte Carlo; weak approximation; error expansion; a posteriori error estimates; adaptivity; error control; mathematical finance (search for similar items in EconPapers)
Date: 2011
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Citations: View citations in EconPapers (1)

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DOI: 10.1515/mcma.2011.003

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