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Options exotiques dans les modèles exponentiels de Lévy

Abstract : The purpose of this thesis is to study the pricing of exotic options in exponential Lévy models. In the first chapter we define Lévy processes, and present their properties. In the second chapter we first recall results on the existence and regularity of density of Lévy processes. For the supremum process of a Lévy process, we give sufficient conditions for existence and regularity of its probability density function (theorems 2.17, 2.19 and 2.23). In the third chapter we study the errors between the supremum of a continuous Lévy process and its discrete version. In the first part of the chapter, we focus on the L1 error (theorems 3.4, 3.8 and 3.11) using a reformulation of Spitzer's identity for Lévy processes (Proposition 3.2). In the second part, we extend the theorem of Asmussen Glynn Pitman to Lévy processes with finite activity and infinite variation (Theorem 3.14). The last part of this chapter examines the specific case of compound Poisson process (Theorem 3.15). In the fourth chapter we focus on the errors of approximation of small jumps of Lévy processes with infinite activity. The second part of the chapter is devoted to the study of the truncation of small jumps. In the third part we study the approximation of small jumps by a Brownian motion using Theorem 4.23, which results from the application of Skorokhod embedding theorem. In the last section, we compare the two approximation methods. In the fifth chapter we apply the results of previous chapters to the pricing of exotic options (barrier, lookback and Asian). We study first the asymptotic behavior of the errors due to discretization. We show that in the case of lookback and barrier options, corrections are possible. The main result for the correction of barrier options is Theorem 5.1. We also study the errors due to the approximation of small jumps. In the last part of this chapter, using Theorem 5.60 and Lemma 5.61 (which is a consequence of Theorem 2.23), we evaluate lookback and digital barrier options (continuous) by semi-closed formulas, assuming that there is no positive jump. Finally, in the sixth chapter we propose Monte Carlo methods to price some exotic options. In the finite activity case, we apply corrections obtained in the fifth chapter, and we use variance reduction techniques. In the infinite activity case, we propose one method to price these options. In the case of Asian options, we provide simple formulas to price these options when the original Lévy process has no Brownian part.
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Contributor : El Hadj Aly Dia <>
Submitted on : Thursday, September 23, 2010 - 4:43:52 PM
Last modification on : Wednesday, November 4, 2020 - 3:36:35 AM
Long-term archiving on: : Thursday, October 25, 2012 - 11:25:22 AM



  • HAL Id : tel-00520583, version 1


El Hadj Aly Dia. Options exotiques dans les modèles exponentiels de Lévy. Mathématiques [math]. Université Paris-Est, 2010. Français. ⟨tel-00520583v1⟩



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