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Modélisation de la dépendance et simulation de processus en finance

Abstract : The first part of this thesis deals with probabilistic numerical methods for simulating the solution of a stochastic differential equation (SDE). We start with the algorithm of Beskos et al. [13] which allows exact simulation of the solution of a one dimensional SDE. We present an extension for the exact computation of expectations and we study the application of these techniques for the pricing of Asian options in the Black & Scholes model. Then, in the second chapter, we propose and study the convergence of two discretization schemes for a family of stochastic volatility models. The first one is well adapted for the pricing of vanilla options and the second one is efficient for the pricing of path-dependent options. We also study the particular case of an Orstein-Uhlenbeck process driving the volatility and we exhibit a third discretization scheme which has better convergence properties. Finally, in the third chapter, we tackle the trajectorial weak convergence of the Euler scheme by providing a simple proof for the estimation of the Wasserstein distance between the solution and its Euler scheme, uniformly in time. The second part of the thesis is dedicated to the modelling of dependence in finance through two examples : the joint modelling of an index together with its composing stocks and intensity-based credit portfolio models. In the forth chapter, we propose a new modelling framework in which the volatility of an index and the volatilities of its composing stocks are connected. When the number of stocks is large, we obtain a simplified model consisting of a local volatility model for the index and a stochastic volatility model for the stocks composed of an intrinsic part and a systemic part driven by the index. We study the calibration of these models and show that it is possible to fit the market prices of both the index and the stocks. Finally, in the last chapter of the thesis, we define an intensity-based credit portfolio model. In order to obtain stronger dependence levels between rating transitions, we extend it by introducing an unobservable random process (frailty) which acts multiplicatively on the intensities of the firms of the portfolio. Our approach is fully historical and we estimate the parameters of our model to past rating transitions using maximum likelihood techniques
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Submitted on : Wednesday, November 24, 2010 - 10:58:22 AM
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Mohamed Sbai Sbaï. Modélisation de la dépendance et simulation de processus en finance. Mathématiques générales [math.GM]. Université Paris-Est, 2009. Français. ⟨NNT : 2009PEST1046⟩. ⟨tel-00451008v2⟩



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