Modélisation de la courbe de variance et modèles à volatilité stochastique

Abstract : The first part of this thesis deals with issues related to the Markov-modeling of the forward variance curve. It is divided into 3 chapters. In the first chapter, we present the general framework of the HJM-type modelling for the forward variance curve. We revisit the Affine-Markov framework, and illustrate by the model proposed by B"uhler 2006. In the second chapter, we propose a new model for the forward variance curve that combines features of the continuous and discrete version of Bergomi's model model Bergomi (2008), without being reduced to either of them. One of the strengths of this model is that the prices of VIX futures and options can be expressed as expectations of deterministic functions of a Gaussian random variable, which reduces the problem of calibration to the inversion of some monotonic functions. In the third chapter, we propose an approximation method for pricing of European options under some lognormal stochastic volatility models (including the model presented in the second chapter, Bergomi's model2008 and Scot model 1987). We obtain an expansion (with respect to the the volatility of volatility parameters of order 3) of the density of the underlying. We also propose a control variate method to effectively reduce variances of Monte Carlo simulations for pricing European optionsThe purpose of the second part of this thesis is to study the monotonicity properties of the prices of European options with respect to the CIR parameters under Heston model. It is divided into two chapters. In the first chapter (see Chapter 4), we give some general results related to the CIR process. We first show that the distribution tails of a combination of the CIR and its arithmetic mean behave as exponential. We then study the derivatives of the solution process with respect to the parameters of its dynamics. These data are derived as solutions of stochastic differential equations, which solves for the representations of these derivatives based on trajectories of the CIR. Chapter 5 is devoted to the study of the monotony of the European price of a put with respect to parameters of CIR and correlation in the Heston model. We show that under certain conditions, prices of European options are monotonic with respect to the parameters of the drift of the CIR. We then show that the parameter of the volatility of volatility plays the role of volatility if we take the realized variance as the underlying. In particular, prices of (convex) options on realized variance are strictly increasing with respect to the volatility of volatility. Finally, we study the monotony of the European Put prices with respect to the correlation. We show that the price of the put is increasing with respect to the correlation for small values ​​of Spot and decreasing for large values. We then study the change points of monotonicity for short and long maturities
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Mathematics [math]. Université Paris-Est, 2011. French. <NNT : 2011PEST1041>


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Sidi Mohamed Ould Aly. Modélisation de la courbe de variance et modèles à volatilité stochastique. Mathematics [math]. Université Paris-Est, 2011. French. <NNT : 2011PEST1041>. <tel-00604530v2>

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