On probability distributions of diffusions and financial models with non-globally smooth coefficients.

Summary Recent work in the field of financial mathematics has highlighted the importance of studying the regularity and fine-grained behavior of distribution tails for certain classes of diffusions with non-globally regular coefficients. In this thesis, we deal with problems arising from this context. We first study the existence, regularity and asymptotics in density space for solutions of stochastic differential equations by imposing only local conditions on the coefficients of the equation. Our analysis is based on the tools of Malliavin calculus and on estimates for Ito processes confined in a tube around a deterministic curve. We obtain significant estimates of the distribution function and density in classes of models including generalizations of the CIR and CEV and models with local-stochastic volatility: in the latter case, the estimates lead to the explosion of the moments of the underlying and thus have an impact on the asymptotic strike behavior of the implied volatility. The parametric modeling of the volatility surface, in turn, is the subject of the second part. We consider the SVI model of J. Gatheral, proposing a new quasi-explicit calibration strategy, whose performance on market data is illustrated. Then, we analyze the ability of SVI to generate approximations for symmetric smiles, by generalizing it to a time-dependent model. We test its application to a Heston model (without and with displacement), generating semi-closed approximations for the volatility smile.