Diffusion approximation of Lévy processes with a view towards finance

Jonas Kiessling 1  and Raúl Tempone 2
  • 1 Institute for Mathematics, Royal Institute of Technology, S-10044 Stockholm, Sweden.
  • 2 Applied Mathematics and Computational Sciences, 4700 King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia.

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.

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This quarterly published journal presents original articles on the theory and applications of Monte Carlo and Quasi-Monte Carlo methods. Launched in 1995 the journal covers all stochastic numerics topics with emphasis on the theory of Monte Carlo methods and new applications in all branches of science and technology.

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