Hostname: page-component-74656f9884-8r8nv Total loading time: 0 Render date: 2026-11-10T03:26:53.495Z Has data issue: false hasContentIssue false
  • English
  • Français

Error Bounds for Small Jumps of Lévy Processes

Part of: Partial differential equations, boundary value problems Markov processes Stochastic processes

Published online by Cambridge University Press:  04 January 2016

E. H. A. Dia
E. H. A. Dia*
Affiliation:
Université Paris-Est
*
∗ Postal address: Laboratoire d'Analyse et de Mathématiques Appliquées, Université Paris-Est, UMR CNRS 8050, 5 boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée, France. Email address: dia.eha@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

The pricing of options in exponential Lévy models amounts to the computation of expectations of functionals of Lévy processes. In many situations, Monte Carlo methods are used. However, the simulation of a Lévy process with infinite Lévy measure generally requires either truncating or replacing the small jumps by a Brownian motion with the same variance. We will derive bounds for the errors generated by these two types of approximation.

Keywords

approximation of small jumpsLévy processSkorokhod embeddingSpitzer identity

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes 65N15: Error bounds
Secondary: 60J75: Jump processes

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 1 , March 2013 , pp. 86 - 105
Copyright
© Applied Probability Trust 

References

Asmussen, S. and Rosinski, J. (2001). Approximations of small Jumps of Lévy processes with a view towards simulation. J. Appl. Prob. 38, 482–493.Google Scholar
Barndorff-Nielsen, O. E. (1997). Normal inverse Gaussian distributions and stochastic volatility modelling. Scand. J. Statist. 24, 1–13.Google Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Broadie, M. and Yamamoto, Y. (2005). A double-exponential fast Gauss transform algorithm for pricing discrete path-dependent options. Operat. Res. 53, 764–779.Google Scholar
Carr, P., Geman, H., Madan, D. B. and Yor, M. (2002). The fine structure of asset returns: an empirical investigation. J. Business. 75, 305–332.Google Scholar
Chaumont, L. (2013). On the law of the supremum of Lévy processes. To appear in Ann. Prob. Google Scholar
Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman & Hall, Boca Raton, FL.Google Scholar
Cont, R. and Voltchkova, E. (2005). Integro-differential equations for option prices in exponential Lévy models. Finance Stoch. 9, 299–325.Google Scholar
Dia, E. H. A. (2010). Options exotiques dans les modèles exponentiels de Lévy. Doctoral Thesis, Université Paris-Est. Available at http://tel.archives-ouvertes.fr/tel-00520583/en/.Google Scholar
Dia, E. H. A. and Lamberton, D. (2011). Connecting discrete and continuous lookback or hindsight options in exponential Lévy models. Adv. Appl. Prob. 43, 1136–1165.Google Scholar
Eberlein, E. (2001). Application of generalized hyperbolic Lévy motions to finance. In Lévy Processes, eds Barndorff-Nielsen, O. E., Mikosch, T. and Resnick, S.. Birkhäuser, Boston, MA, pp. 319–337.Google Scholar
Feng, L. and Linetsky, V. (2008). Pricing discretely monitored barrier options and defaultable bonds in Lévy process models: a fast Hilbert transform approach. Math. Finance 18, 337–384.Google Scholar
Feng, L. and Linetsky, V. (2009). Computing exponential moments of the discrete maximum of a Lévy process and lookback options. Finance Stoch. 13, 501–529.Google Scholar
Morris, C. N. (1982). Natural exponential families with quadratic variance functions. Ann. Statist. 10, 65–80.Google Scholar
Petrella, G. and Kou, S. G. (2004). Numerical pricing of discrete barrier and lookback options via Laplace transforms. J. Comput. Finance 8, 1–37.Google Scholar
Rydberg, T. H. (1997). The normal inverse Gaussian Lévy process: simulation and approximation. Commun. Statist. Stoch. Models 13, 887–910.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Signahl, M. (2003). On error rates in normal approximations and simulation schemes for Lévy processes. Stoch. Models 19, 287–298.Google Scholar
Skorokhod, A. V. (1965). Studies in the Theory of Random Processes. Addison-Wesley, Reading, MA.Google Scholar
Tankov, P. (2004). Lévy processes in finance: inverse problems and dependence modelling. Doctoral Thesis, Ecole Polytechnique.Google Scholar