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

  • Hélène Guérin (a1) and Jean-François Renaud (a2)

We study the distribution Ex[exp(-q0t1(a,b)(Xs)ds); Xt ∈ dy], where -∞ ≤ a < b < ∞, and where q, t > 0 and xR 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.

Corresponding author
* Postal address: IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France. Email address:
** Postal address: Département de Mathématiques, Université du Québec à Montréal, 201 av. Président-Kennedy, Montréal, QC H2X 3Y7, Canada. Email address:
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[1]Akahori, J. (1995). Some formulae for a new type of path-dependent option. Ann. Appl. Prob. 5, 383388.
[2]Bertoin, J. (1996). Lévy Processes. Cambridge University Press.
[3]Cai, N. (2009). On first passage times of a hyper-exponential jump diffusion process. Operat. Res. Lett. 37, 127134.
[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, 412437.
[5]Hugonnier, J.-N. (1999). The Feynman–Kac formula and pricing occupation time derivatives. Internat. J. Theoret. Appl. Finance 2, 153178.
[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. 97186.
[7]Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
[8]Kyprianou, A. E., Pardo, J. C. and Pérez, J. L. (2014). Occupation times of refracted Lévy processes. J. Theoret. Prob. 27, 12921315.
[9]Lachal, A. (2013). Sojourn time in an union of intervals for diffusions. Methodology Comput. Appl. Prob. 15, 743771.
[10]Landriault, D., Renaud, J.-F. and Zhou, X. (2011). Occupation times of spectrally negative Lévy processes with applications. Stoch. Process. Appl. 121, 26292641.
[11]Landriault, D., Renaud, J.-F. and Zhou, X. (2014). An insurance risk model with Parisian implementation delays. Methodology Comput. Appl. Prob. 16, 583607.
[12]Li, B. and Zhou, X. (2013). The joint Laplace transforms for diffusion occupation times. Adv. Appl. Prob. 45, 10491067.
[13]Linetsky, V. (1999). Step options. Math. Finance 9, 5596.
[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, 14081435.
[15]Renaud, J.-F. (2014). On the time spent in the red by a refracted Lévy risk process. J. Appl. Prob. 51, 11711188.
[16]Takács, L. (1996). On a generalization of the arc-sine law. Ann. Appl. Prob. 6, 10351040.
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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
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