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

Part of: Stochastic processes Mathematical finance

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: [email protected]
** Postal address: Département de Mathématiques, Université du Québec à Montréal, 201 av. Président-Kennedy, Montréal, QC H2X 3Y7, Canada. Email address: [email protected]
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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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References

[1]Akahori, J. (1995). Some formulae for a new type of path-dependent option. Ann. Appl. Prob. 5, 383–388. CrossRefGoogle Scholar
[2]Bertoin, J. (1996). Lévy Processes. Cambridge University Press. Google Scholar
[3]Cai, N. (2009). On first passage times of a hyper-exponential jump diffusion process. Operat. Res. Lett. 37, 127–134. CrossRefGoogle Scholar
[4]Cai, N., Chen, N. and Wan, X. (2010). Occupation times of jump-diffusion processes with double exponential jumps and the pricing of options. Math. Operat. Res. 35, 412–437. CrossRefGoogle Scholar
[5]Hugonnier, J.-N. (1999). The Feynman–Kac formula and pricing occupation time derivatives. Internat. J. Theoret. Appl. Finance 2, 153–178. Google Scholar
[6]Kuznetsov, A., Kyprianou, A. E. and Rivero, V. (2012). The theory of scale functions for spectrally negative Lévy processes. In Lévy Matters (Lecture Notes Math. 2061), Springer, Heidelberg, pp. 97–186. Google Scholar
[7]Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin. Google Scholar
[8]Kyprianou, A. E., Pardo, J. C. and Pérez, J. L. (2014). Occupation times of refracted Lévy processes. J. Theoret. Prob. 27, 1292–1315. Google Scholar
[9]Lachal, A. (2013). Sojourn time in an union of intervals for diffusions. Methodology Comput. Appl. Prob. 15, 743–771. CrossRefGoogle Scholar
[10]Landriault, D., Renaud, J.-F. and Zhou, X. (2011). Occupation times of spectrally negative Lévy processes with applications. Stoch. Process. Appl. 121, 2629–2641. Google Scholar
[11]Landriault, D., Renaud, J.-F. and Zhou, X. (2014). An insurance risk model with Parisian implementation delays. Methodology Comput. Appl. Prob. 16, 583–607. Google Scholar
[12]Li, B. and Zhou, X. (2013). The joint Laplace transforms for diffusion occupation times. Adv. Appl. Prob. 45, 1049–1067. CrossRefGoogle Scholar
[13]Linetsky, V. (1999). Step options. Math. Finance 9, 55–96. Google Scholar
[14]Loeffen, R. L., Renaud, J.-F. and Zhou, X. (2014). Occupation times of intervals until first passage times for spectrally negative Lévy processes. Stoch. Process. Appl. 124, 1408–1435. Google Scholar
[15]Renaud, J.-F. (2014). On the time spent in the red by a refracted Lévy risk process. J. Appl. Prob. 51, 1171–1188. Google Scholar
[16]Takács, L. (1996). On a generalization of the arc-sine law. Ann. Appl. Prob. 6, 1035–1040. Google Scholar