| In this thesis we study the pricing of single-name and multi-name credit derivatives and analyze it with asymptotic methods for the solution of partial differential equations. We employ the intensity-based framework for the modeling of the arrival of a default. The first part of the thesis describes the application of regular and singular perturbation expansions on the intensity of default from which we derive approximations for the pricing functions of several defaultable derivatives, such as defaultable bonds, options on defaultable bonds, and credit default swaps. In particular, we assume an Ornstein-Uhlenbeck process for the interest rate, and a two-factor diffusion model for the intensity of default. The approximation allows for computational efficiency in calibrating the model. Empirical evidence validating the existence of multiple scales is presented by the calibration of the model on corporate yield curves. In the second part, we examine the most popular type of multi-name credit derivative: collateralized debt obligations (CDOs). The pricing of CDOs and other basket credit derivatives is contingent upon (i) a realistic modeling of the firms' default times and the correlation between them, and (ii) efficient computational methods for computing the portfolio loss distribution from the firms' marginal default time distributions. Factor models, a widely-used class of pricing models in the literature, are computationally tractable despite the large dimension of the pricing problem, thus satisfying issue (ii), but they are less accurate in the assessment of default risk in the pool of firms (issue (i)). We revisit the intensity-based modeling setup for basket credit derivatives, and with the aforementioned issues in mind we propose improvements (a) via incorporating fast mean-reverting stochastic volatility in the default intensity processes, and (b) by breaking firms into homogeneous groups within the original set of firms. We present a calibration example, and discuss the relative performance of the approach. |
