Valuation of portfolio credit derivatives and data - based default prediction

To my parents and to Diana Preface Credit derivatives play a major role in financial markets. To price these complex products objectively advanced mathematical models are needed. In this thesis we focus on the modelling and valuation of two portfolio credit derivatives: n th-to-default credit default swaps (CDSs) and collateralised debt obligations (CDOs). The key input parameters of these models are the default probabilities of individual portfolio components. It is an insight of the recent financial crisis that failure in rating credits correctly leads to severe mispricing of risky credit products. Therefore we concentrate also on the development of an improved default prediction model. This thesis is divided into two parts. In the first one, that exists of the Chapters 1–5, we study the pricing of n th-to-default credit default swaps and collateralised debt obligations. It is a continuation of the research that started with the author's diploma thesis (Pohl (2006)). We extend the valuation of these contracts to portfolios with individual nominal amounts and individual loss-given-defaults. Furthermore we explore an extension for individual stochastic loss-given-defaults. Besides the derivation of pricing formulas in that general setting the emphasis in this part of the investigation is also on the numerical computation. In particular we make a sensitivity analysis for various model parameters. The first part of the thesis is structured as follows. In Chapter 1 we present the factor model approach that is used to model the dependence structure in the underlying reference portfolio and a model for stochastic loss-given-default. In Chapter 2 we present a recursive procedure that allows the calculation of the distribution function of the time of the n th default in the reference portfolio at any point in time and analyse some of its features. This function is the basis for the valuation of n th-to-default CDSs on portfolios with homogeneous nominal amounts and homogeneous deterministic loss-given-defaults. We adjust this procedure to the calculation of the distribution function of the time of the n th default caused by a certain name and its conditional counterpart given the systematic factor which allows the valuation of these contracts in a general setting with individual nominal amounts and individual stochastic loss-given-defaults. In Chapter 3 we develop an advanced method for the efficient computation of the quantiles of a sum of two independent normal inverse Gaussian distributed random variables which is required if normal inverse Gaussian distributed variables are used in …