In the first part of the work, a survey of the credit risk literature is given, which offers a quick introduction into the area and presents the mathematical methods in a unifying way. Second, we propose two new models of credit risk, focusing on different needs. The first model generalizes existing models using random fields in Hilbert spaces. The second model uses Gaussian random fields leading to explicit formulas for a number of derivatives, for which we propose two calibration procedures.
The work is organized as follows. In Chapter 1, a survey of the credit risk literature is given. This includes structural models, hazard rate models, methods incorporating credit ratings, models for baskets of credit risky bonds, hybrid models, market models and commercial models. In the last section we illustrate several credit derivatives. Generally the mathematical framework for the models is provided and some models are discussed in greater detail. Additionally, an explicit formula for the default intensity in the imperfect information model of Duffie and Lando (2001) is derived.
Chapters 2 and 3 focus on credit risk modeling using stochastic differential equations (SDEs) in infinite dimensions. Although known in interest rate theory, the application of these methods is new to credit risk. Chapter 2 contains an introduction to SDEs in Hilbert spaces providing an Ito formula which is adequate for our purposes. In Chapter 3 a Heath-Jarrow-Morton formulation of credit risk in infinite dimensions is given. The work of Duffie and Singleton (1999) and Bielecki and Rutkowski (2000) was enhanced with alternative recovery models and extended to infinite dimensions. These new models comprise most of the known credit risk models and still offer frameworks which are tractable.
In Chapter 4, a credit risk model is presented which uses Gaussian random fields and transfers the framework of Kennedy (1994) to credit risk. In contrast to the functional analytic approach in the previous two chapters, the methods used in this section concentrate on deriving formulas for pricing and hedging. Explicit expressions for the prices of several credit default options are obtained and an example for hedging credit derivatives is presented.
Based on these pricing formulas, two calibration methodologies are provided. The first calibration procedure fits the model to prices of derivatives using a least squares approach. As the data for derivatives like credit default swaptions is still scarce, the second approach takes this into account and in addition uses historical data. This new approach allows to calibrate perfectly to market prices and is applicable using only a small amount of credit derivatives data.
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