Valuation of Xva adjustments: from expected exposure to adverse correlation risks.

Summary We begin this thesis report by evaluating the expected exposure, which represents one of the major components of XVA adjustments. Under the assumption of independence between exposure and financing and credit costs, we derive in Chapter 3 a new representation of expected exposure as the solution of an ordinary differential equation with respect to the time of default observation. For the one-dimensional case, we rely on arguments similar to those for Dupire's local volatility. And for the multidimensional case, we refer to the Co-aire formula. This representation allows us to explain the impact of volatility on the expected exposure: this time value involves the volatility of the underlyings as well as the first-order sensitivity of the price, evaluated on a finite set of points. Despite numerical limitations, this method is an accurate and fast approach for valuing unit XVA in dimension 1 and 2.The following chapters are dedicated to the risk aspects of correlations between exposure envelopes and XVA costs. We present a model of the general correlation risk through a multivariate stochastic diffusion, including both the underlying assets of the derivatives and the default intensities. In this framework, we present a new approach to valuation by asymptotic developments, such that the price of an XVA adjustment corresponds to the price of the zero-correlation adjustment, plus an explicit sum of corrective terms. Chapter 4 is devoted to the technical derivation and study of the numerical error in the context of the valuation of default contingent derivatives. The quality of the numerical approximations depends solely on the regularity of the credit intensity diffusion process, and is independent of the regularity of the payoff function. The valuation formulas for CVA and FVA are presented in Chapter 5. A generalization of the asymptotic developments for the bilateral default framework is addressed in Chapter 6.We conclude this dissertation by addressing a case of the specific correlation risk related to rating migration contracts. Beyond the valuation formulas, our contribution consists in presenting a robust approach for the construction and calibration of a rating transition model consistent with market implied default probabilities.