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Joint distribution of a spectrally negative Lévy process and its occupation time, with step option pricing in view

Part of: Mathematical finance Stochastic processes

Published online by Cambridge University Press:  24 March 2016

Hélène Guérin  and
Jean-François Renaud
Hélène Guérin*
Affiliation:
Université de Rennes 1
Jean-François Renaud*
Affiliation:
Université du Québec à Montréal (UQAM)
*
* Postal address: IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France. Email address: helene.guerin@univ-rennes1.fr
** Postal address: Département de Mathématiques, Université du Québec à Montréal, 201 av. Président-Kennedy, Montréal, QC H2X 3Y7, Canada. Email address: renaud.jf@uqam.ca
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Abstract

We study the distribution Ex[exp(-q∫0t1(a,b)(Xs)ds); Xt ∈ dy], where -∞ ≤ a < b < ∞, and where q, t > 0 and x ∈ R for a spectrally negative Lévy process X. More precisely, we identify the Laplace transform with respect to t of this measure in terms of the scale functions of the underlying process. Our results are then used to price step options and the particular case of an exponential spectrally negative Lévy jump-diffusion model is discussed.

Keywords

Occupation timespectrally negative Lévy processfluctuation theoryscale functionstep option

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 91G20: Derivative securities
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 1 , March 2016 , pp. 274 - 297
Copyright
Copyright © Applied Probability Trust 2016 

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