Étude asymptotique des algorithmes stochastiques et calcul du prix des options parisiennes

Abstract : This thesis is split into two parts. The first one deals with the study of stochastic algorithms. In an introductory chapter, we present the [55] algorithm while making a parallel with the Newton algorithm commonly used in deterministic optimisation problems. These reminders naturally lead to the presentation of randomly truncated stochastic algorithms as first introduced by [21]. The first study of these randomly truncated stochastic algorithms is concerned with their almost sure convergence which has already been established under varying hypotheses. The first chapter gives us the opportunity to try to clarify the assumptions a little and to present a simplified proof of the almost sure convergence. The second chapter is devoted to the study of the convergence rate. More precisely, we consider a moving window version of the algorithm and establish a central limit theorem. The last chapter of this first part presents two applications of stochastic algorithms to finance. The first one deals with the calibration of the correlation in a multidimensional market model, while the second one is based on the work of [7]. Meanwhile, we improve the results Arouna had obtained. The second part of the thesis is concerned with the pricing of Parisian options. The valuation technique is based on computing closed form formula for the Laplace transforms of the prices following the seminar work of Chesney, Jeanblanc-Picqué, and Yor [23] on the topic. First, we determine these formulae for the single barrier Parisian options following closely [23], second we do the same for double barrier Parisian options. Then, we study the numerical inversion of these Laplace transforms based on a contour integral technique. We establish the accuracy of the method we use. To do so, we prove the regularity of the Parisian option prices and establish the existence of a density with respect to the Lebesgue measure for the
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Mathématiques [math]. Ecole des Ponts ParisTech, 2007. Français. 〈NNT : 2007ENPC0714〉
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Jérôme Lelong. Étude asymptotique des algorithmes stochastiques et calcul du prix des options parisiennes. Mathématiques [math]. Ecole des Ponts ParisTech, 2007. Français. 〈NNT : 2007ENPC0714〉. 〈pastel-00003310〉



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