Stochastic expansion for the diffusion processes and applications to option pricing.

Summary This thesis is devoted to the approximation of the expectation of a functional (which can depend on the whole trajectory) applied to a diffusion process (which can be multidimensional). The motivation for this work comes from financial mathematics where the valuation of options is reduced to the calculation of such expectations. The speed of the price calculations and calibration procedures is a very strong operational constraint and we bring real-time tools (or at least more competitive than Monte Carlo simulations in the multidimensional case) to meet these needs. To obtain approximation formulas, we choose a proxy model in which analytical calculations are possible, then we use stochastic developments around this proxy model and the Malliavin calculation to approximate the quantities of interest. In the case where the Malliavin calculus cannot be applied, we develop an alternative methodology combining Itô calculus and PDE arguments. All approaches (ranging from PDEs to stochastic analysis) provide explicit formulas and accurate error estimates as a function of the model parameters. Although the end result is often the same, the explicit derivation of the development can be very different and we compare the approaches, both in terms of how the correction terms are made explicit and the assumptions required to obtain the error estimates. We consider different classes of models and functionals in the four parts of the thesis. In Part I, we focus on local volatility models and obtain new approximation formulas for prices, sensitivities (delta) and implied volatilities of vanilla products that surpass in accuracy the previously known formulas. We also present new results for the valuation of forward-looking options. Part II deals with the analytical approximation of vanilla prices in models combining local and stochastic volatility (Heston type). This model is very delicate to analyze because its moments are not all finite and 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 on a skillfully modified model, regular in the Malliavin sense and from which we can control the distance from the initial model. Part III deals with the valuation of regular barrier options in the context of local volatility models. This is a case not considered in the literature, difficult because of the exit time indicator. We mix Itô's calculation, PDE arguments, martingale properties and time convolutions of densities in order to decompose the approximation error and to explain the corrective terms. We obtain explicit and very accurate approximation formulas under a martingale assumption. Part IV presents a new methodology (denoted SAFE) for the effective law approximation of multidimensional diffusions in a rather 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 by finite elements. We give an estimate of the complexity of our methodology. We show an improved efficiency compared to Monte Carlo simulations in small and medium dimensions (up to 10).