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Diffusion approximation of Lévy processes with a view towards finance

Author

Listed:
  • 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.)

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.

Suggested Citation

  • Kiessling Jonas & Tempone Raúl, 2011. "Diffusion approximation of Lévy processes with a view towards finance," Monte Carlo Methods and Applications, De Gruyter, vol. 17(1), pages 11-45, January.
    • Handle: RePEc:bpj:mcmeap:v:17:y:2011:i:1:p:11-45:n:3
    DOI: 10.1515/mcma.2011.003
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    References listed on IDEAS

    as
    1. Dilip B. Madan & Peter P. Carr & Eric C. Chang, 1998. "The Variance Gamma Process and Option Pricing," Review of Finance, European Finance Association, vol. 2(1), pages 79-105.
    2. Peter Carr & Hélyette Geman & Dilip B. Madan & Marc Yor, 2003. "Stochastic Volatility for Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 13(3), pages 345-382, July.
    3. repec:dau:papers:123456789/1392 is not listed on IDEAS
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    Cited by:

    1. Laetitia Badouraly Kassim & Jérôme Lelong & Imane Loumrhari, 2015. "Importance sampling for jump processes and applications to finance," Post-Print hal-00842362, HAL.

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