Credit Risk Management and Jump Models

This doctoral thesis comprises three research papers that seek to improve and create corporate and sovereign credit risk models, to provide an approximate analytic expressions for CDS spreads and a numerical method for partial differential equation arisen from pricing defaultable coupon bond. First, an extension of Jump to Default Constant Elasticity Variance in more general and realistic framework is provided (see Chapter 3). We incorporate, in the model introduced in [9], a stochastic interest rate with possible negative values. In addition we provide an asymptotic approximation formula for CDS spreads based on perturbation theory. The robustness and efficiency of the method is conformed by several calibration tests on real market data. Next, under the model introduced in Chapter 3, we present in Chapter 4 a new numerical method for pricing non callable defaultable bond. we propose appropriate numerical schemes based on a Crank-Nicolson semi-Lagrangian method for time discretization combined with biquadratic Lagrange finite elements for space discretization. Once the numerical solutions of the PDEs are obtained, a post-processing procedure is carried out in order to achieve the value of the bond. Finally, we introduce a hybrid Sovereign credit risk model in which the intensity of default of a sovereign is based on the jump to default extended CEV model (see Chapter 5). The model captures the interrelationship between creditworthiness of a sovereign, its intensity to default and the correlation with the exchange rate between the bond's currency and the currency in which the CDS spread are quoted. We consider the Sovereign Credit Default Swaps Italy, during and after the financial crisis, as a case of study to show the effectiveness of our model.