On credit risk modeling and credit derivatives pricing

Abstract

In this thesis, efforts are devoted to the stochastic modeling, measurement and evaluation of credit risks, the development of mathematical and statistical tools to estimate and predict these risks, and methods for solving the significant computational problems arising in this context.

The reduced-form intensity based credit risk models are studied. A new type of reduced-form intensity-based model is introduced, which can incorporate the impacts of both observable trigger events and economic environment on corporate defaults. The key idea of the model is to augment a Cox process with trigger events. In addition, this thesis focuses on the relationship between structural firm value model and reduced-form intensity based model. A continuous time structural asset value model for the asset value of two correlated firms with a two-dimensional Brownian motion is studied. With the incomplete information introduced, the information set available to the market participants includes the default time of each firm and the periodic asset value reports. The original structural model is first transformed into a reduced-form model. Then the conditional distribution of the default time as well as the asset value of each name are derived. The existence of the intensity processes of default times is proven and explicit form of intensity processes is given in this thesis.

Discrete-time Markovian models in credit crisis are considered. Markovian models are proposed to capture the default correlation in a multi-sector economy. The main idea is to describe the infection (defaults) in various sectors by using an epidemic model. Green’s model, an epidemic model, is applied to characterize the infectious effect in each sector and dependence structures among various sectors are also proposed. The models are then applied to the computation of Crisis Value-at-Risk (CVaR) and Crisis Expected Shortfall (CES). The relationship between correlated defaults of different industrial sectors and business cycles as well as the impacts of business cycles on modeling and predicting correlated defaults is investigated using the Probabilistic Boolean Network (PBN). The idea is to model the credit default process by a PBN and the network structure can be inferred by using Markov chain theory and real-world data.

A reduced-form model for economic and recorded default times is proposed and the probability distributions of these two default times are derived. The numerical study on the difference between these two shows that our proposed model can both capture the features and fit the empirical data. A simple and efficient method, based on the ordered default rate, is derived to compute the ordered default time distributions in both the homogeneous case and the two-group heterogeneous case under the interacting intensity default contagion model. Analytical expressions for the ordered default time distributions with recursive formulas for the coefficients are given, which makes the calculation fast and efficient in finding rates of basket CDSs.