Stochastic expansion for the diffusion processes and applications to option pricing

This thesis deals with the approximation of the expectation of a functional (possibly depending on the whole path) applied to a diffusion process (possibly multidimensional). The motivation for this work comes from financial mathematics where the pricing of options is reduced to the calculation of such expectations. The rapidity for price computations and calibration procedures is a very strong operational constraint and we provide real-time tools (or at least more competitive than Monte Carlo simulations in the case of multidimensional diffusions) to meet these needs. In order to derive approximation formulas, we choose a proxy model in which analytical calculus are possible and then we use stochastic expansions around the proxy model and Malliavin calculus to approach the quantities of interest. In situation where Malliavin calculus can not be applied, we develop an alternative methodology combining Ito calculus and PDE arguments. All the approaches (from PDEs to stochastic analysis) allow to obtain explicit formulas and tight error estimates in terms of the model parameters. Although the final result is generally the same, the derivation can be quite different and we compare the approaches, first regarding the way in which the corrective terms are made explicit, second regarding the error estimates and the assumptions used for that. We consider various classes of models and functionals throughout the four Parts of the thesis. In the Part I, we focus on local volatility models and provide new price, sensitivity (delta) and implied volatility approximation formulas for vanilla products showing an improving accuracy in comparison to previous known formulas. We also introduce new results concerning the pricing of forward start options. The Part II deals with the analytical approximation of vanilla prices in models combining both local and stochastic volatility (Heston type). This model is very difficult to analyze because its moments can explode and because it is not regular in the Malliavin sense. The error analysis is original and the idea is to work on an appropriate regularization of the payoff and a suitably perturbed model, regular in the Malliavin sense and from which the distance with the initial model can be controlled. The Part III covers the pricing of regular barrier options in the framework of local volatility models. This is a difficult issue due to the indicator function on the exit times which is not considered in the literature. We use an approach mixing Ito calculus, PDE arguments, martingale properties and temporal convolutions of densities to decompose the approximation error and to compute correction terms. We obtain explicit and accurate approximation formulas under a martingale hypothesis. The Part IV introduces a new methodology (denoted by SAFE) for the efficient weak analytical approximation of multidimensional diffusions in a quite general framework. We combine the use of a Gaussian proxy to approximate the law of the multidimensional diffusion and a local interpolation of the terminal function using Finite Elements. We give estimates of the complexity of our methodology. We show an improved efficiency in comparison to Monte Carlo simulations in small and medium dimensions (up to 10).