Advertisement

Lithuanian Mathematical Journal

, Volume 55, Issue 2, pp 222–230 | Cite as

Small-Maturity Digital Options in Lévy Models: An Analytic Approach*

  • Stefan GerholdEmail author
Article

Abstract

We prove a small-time Tauberian theorem for transition probabilities of certain Lévy processes. The main assumption is a condition on the asymptotic behavior of the characteristic function. This gives an alternative derivation of some results on digital options and implied volatility slopes in Lévy models. In probabilistic terms, it gives a sufficient criterion for Spitzer’s condition.

Keywords

Lévy process characteristic function asymptotics option pricing 

MSC

60G51 60E10 91G20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R.A. Doney, Fluctuation theory for Lévy processes. École d’Été de Probabilités de Saint-Flour XXXV-2005, Lect. Notes Math., Vol. 1897, Springer, Berlin.Google Scholar
  2. 2.
    J.E. Figueroa-López and C. Houdré, Small-time expansions for the transition distributions of Lévy processes, Stochastic Processes Appl., 119(11):3862–3889, 2009, available from: http://dx.doi.org/10.1016/j.spa.2009.09.002 .CrossRefzbMATHGoogle Scholar
  3. 3.
    S. Gerhold and I.C. Gülüm, Small-maturity asymptotics for the at-the-money implied volatility slope in Lévy models, preprint, 2014.Google Scholar
  4. 4.
    R.W. Lee, Option pricing by transform methods: Extensions, unification, and error control, J. Comput. Finance, 7(3):51–86, 2004.Google Scholar
  5. 5.
    M. Rosenbaum and P. Tankov, Asymptotic results for time-changed Lévy processes sampled at hitting times, Stochastic Processes Appl., 121(7):1607–1632, 2011, available from: http://dx.doi.org/10.1016/j.spa.2011.03.013 .CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    W. Schoutens, Meixner processes: Theory and applications in finance, EURANDOM Report 2002-004, EURANDOM, Eindhoven, 2002.Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Vienna University of TechnologyViennaAustria

Personalised recommendations