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Equations integro-differentielles d'évolution: méthodes numériques et applications en finance.

Abstract : This thesis deals with option pricing in exponential Lévy models. We establish the relationship between option prices in such models and partial integro-differential equations (PIDE). This allows us to construct efficient numerical methods for option pricing. First, we study the regularity of European (vanilla or barrier) option prices. In particular, we show on several examples that this regularity may fail. In this case, option prices must be considered as generalized solutions of the PIDEs. More precisely, we prove that European option prices, with or without barriers, are viscosity solutions of the corresponding integro-differential problems. Next, we propose two semi-implicit finite difference schemes for numerical solution of the PIDEs. We study their consistency, stability, and convergence to the viscosity solution of the equation. We also estimate the convergence rate. The last part of the thesis is devoted to numerical tests and comparison of the efficiency of the two schemes proposed.
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Submitted on : Tuesday, July 27, 2010 - 3:44:27 PM
Last modification on : Wednesday, March 27, 2019 - 4:08:30 PM
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Ekaterina Voltchkova. Equations integro-differentielles d'évolution: méthodes numériques et applications en finance.. Mathématiques générales [math.GM]. Ecole Polytechnique X, 2005. Français. ⟨pastel-00001538⟩

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