Dynamic Collateralized Debt Obligations Modeling

Collateralized Debt Obligation (CDO) is an encompassing term that stands for securities backed by a pool of reference entities such as bonds, loans, funds, credit default swaps, and CDO tranches themselves. The origination and sale of CDOs has become one of the most important innovative business lines for the banking industry for the last decade. It comes clear that the modeling and valuation of CDOs plays a crucial role in understanding the mechanics of the current financial crisis and helping to avoid future crises. The main objective of this project is to develop more suitable and efficient valuation models for CDO securities based on a valuation framework for synthetic CDOs on corporate credit indices, such as iTraxx and CDX, which are the most liquid instruments in the CDO market. Point of departure is the valuation framework that has recently been proposed in a paper of Filipovic, Overbeck and Schmidt, "Dynamic CDO Term Structure Modelling"(2008), "FOS" herein. Focus is on the normalized aggregate loss process in the underlying pool of credit risky reference entities. A defaultable (T,x)-bond is defined as security which pays one if the loss process has not exceeded level x at maturity T, and zero else. (T,x)-bonds turn out to be the fundamental basis components for the hedging and pricing of any CDO derivative. They can be factorized into their default and market ("forward spread") risk components. This representation corresponds to a stylized fact of financial markets: spread risk is what primarily drives CDO values; the objective default risk is secondary. The forward spread surface movements can be exogenously specified, taking account of contagion effects from the loss process. Necessary and sufficient conditions for the absence of arbitrage lead to a non-classical stochastic differential equation ("FOS equation" herein) for the forward spreads. This project can roughly be divided into two interacting tracks: a more mathematical-formal and a more empirical one. The first track investigates the performance of the very generally formulated dynamic FOS framework in different concrete settings specifications, and extends and fills some remaining theoretical gaps in the framework. In particular, it elaborates on research questions such as stochastic simulation and valuation of the FOS equation, finite dimensional state space models, relation of the top-down model to the constituent credit default swaps, and the development of appropriate market models. The second track bridges the gap between theory and empirics. It is going to be particularly data use intensive. Focus is on the calibration of the forward rate surface to market data, estimating the volatility and drift of the forward rates, estimating the density between actual measure P and risk- neutral measure Q, and estimating contagion effects. A secondary aim would be the use of the calibration results to make economic factor analysis, parameterizations and diagnostics of the risk aversion, and to formulate and quantify trading strategies based on the instruments used. Further, the factor analysis can help us to get a better picture of the evolution of the credit cycle and its connection to the business cycle. Risk management techniques will be addressed too.

The underlying focus of this FWF project has been the modelling of joint default risk across different constituents of a pool of credit risky assets in the context of collateralized debt obligations. Methodologically, this task boils down, firstly, to modelling the dependence structure of a set of potentially connected portfolios or institutions, and, secondly, to the estimation of model parameters describing expected return and risk of the underlying financial variables. This aspect has become particularly important in light of the recent financial crisis where the meltdown of the market for collateralized debt obligations the main reason for the breakout of the crisis. Following the recent attempts to explain the crisis we find that there was and still is a lack of understanding the effects of connectivity among financial institutions (or the network effects between them) and a lack of appropriate estimation techniques for key risk and return parameters which are necessary for any financial decision making.As a consequence, one of the work streams of this project has changed its focus away from collateralized debt obligations as the primary object of study towards the financial system represented as a stochastic network of financial institutions (banks). Indeed, over the last decade, systemic risk in financial networks has emerged as a major research topic in mathematical finance. Common theme is the modelling and measuring of spill-over effects to capture externalities that an individual institution imposes on the financial system. This includes default contagion in particular. We show that random graph theory and stochastic networks, which have previously been studied in the context of telecommunications, epidemics and percolation and other fields, provide an appropriate tool kit for capturing the aspects of financial systemic risk.A second work stream was the application and extension of appropriate estimation techniques for models of joint risk processes in the context of collateralized debt obligations and other financial markets. We show that the use of Bayesian estimation techniques (e.g., predictive systems or Markov Chain Monte Carlo methods) prove to be superior to traditional methods (e.g., predictive regression or Kalman filter). We find that the use of these techniques not only improves the quality of financial decision making but also contributes significantly to the work of regulators and policy makers.