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Modèles de Lévy exponentiels en finance : mesures de f-divergence minimale et modèles avec change-point

Abstract : In this thesis, we study various aspects of exponential Levy models in finance, and in particular : 1. continuity properties for option prices as functions of the Levy process's parameters, 2. the preservation of theLevy property under f-divergence minimal martingale measures, 3. change-point models, obtained by switching from the exponential of one Levy process to another at a random time. In order to obtain continuity properties, we first give convergence results for Levy processes under martingale measures. The convergence of a number of option prices then follows from the Wiener-Hopf factorisation. We also give some continuity results for option prices under various f-divergence minimal martingale measures. It has been noted that the Levy property is preserved by all minimal measures associated with an f-divergence whose second derivative is a power function. We show that under some conditions on the parameters of the Levy process, this preservation property only holds for these classical f-divergences. The duality between utility maximisation and f-divergence minimisation then enables us to obtain a general formula for some optimal strategies. In our study of change-point models, we describe the form of f-minimal martingale measures, in particular in relation to the minimal measures associated with the two underlying Levy processes. We also give an expression of some utility maximising optimal strategies.
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Submitted on : Friday, April 1, 2011 - 11:31:40 AM
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  • HAL Id : tel-00582378, version 1


Suzanne Cawston. Modèles de Lévy exponentiels en finance : mesures de f-divergence minimale et modèles avec change-point. Mathématiques [math]. Université d'Angers, 2010. Français. ⟨tel-00582378⟩



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