On Probability Distributions of Diffusions and Financial Models with non-globally smooth coefficients

In this Ph.D. dissertation we deal with the issue of the regularity and the estimation of probability laws for diffusions with non-globally smooth coefficients, with particular focus on financial models. The analysis of probability laws for the solutions of Stochastic Differential Equations (SDEs) driven by the Brownian motion is among the main applications of the Malliavin calculus on the Wiener space: typical issues involve the existence and smoothness of a density, and the study of the asymptotic behaviour of the distribution’s tails. The classical results in this area are stated assuming global regularity conditions on the coefficients of the SDE: an assumption which fails to be fulfilled by several financial models, whose coefficients involve square-root or other non-Lipschitz continuous functions. Then, in the first part of this thesis (chapters 2, 3 and 4) we study the existence, smoothness and space asymptotics of densities when only local conditions on the coefficients of the SDE are considered. Our analysis is based on Malliavin calculus tools and on tube estimates for Itô processes, namely estimates on the probability that an Itô process remains around a deterministic curve up to a given time. We give applications of our results to general classes of option pricing models, including generalisations of CIR and CEV processes and Local Stochastic Volatility models. In the latter case, the estimates we derive on the law of the underlying price have an impact on moment explosion and, consequently, on the large-strike asymptotic behaviour of the implied volatility. Implied volatility modeling, in its turn, makes the object of the second part of this thesis (chapters 5 and 6). We deal with some questions related to the issue of an efficient and economical parametric modelisation of the volatility surface. We focus on J. Gatheral’s SVI model, first tackling the problem of its calibration to the market smile. We propose an effective quasi-explicit calibration procedure and display its performances on financial data. Then, we analyse the capability of SVI to generate efficient time-dependent approximations of symmetric smiles, providing the corresponding numerical applications in the framework of the Heston stochastic volatility model.