Pricing models for collateralized debt obligations

One of the most controversial and innovative financial products in recent years has been collateralised debt obligations (CDOs). Much of the blame for the current credit crisis is being attributed to the mathematical models and quantitative methods associated with these credit derivatives. In recent years, there has been rapidly growing research on credit derivatives and correlated defaults and in this thesis we examine the possible replacement of current copula based approaches with intuitive contagion models for percolation on finite networks. We propose that modelling the probability of default in a correlated portfolio is similar to modelling the probability of default of contagion spreading in a network. In the first part of our thesis we review current models from the literature that have been suggested to price CDOs. From the literature review we noted that the critical input in the pricing of a CDO is an estimate of the default dependence (default correlation) between the underlying names in a portfolio. Dependency modelling with copula functions, introduced by Li (2000), has become a market standard in the pricing of CDOs. We compute the default distribution for both the Gaussian and Student t4 copulas by implementing both a Monte Carlo and theoretical approach. We then compare these copulas ability to fit market data. In the second part of our thesis we begin by introducing some of the earliest network models proposed by Paul Erdős and Alfred Renyi. We show how similar to component sizes on a network being related to contagion on a network, it is possible then to compute the probability of default for a portfolio of names using network percolation. We present the finite and infinite network models used to model the default dependency between names in a portfolio and thus extend the infinite network model to compute the probability of default for a CDO portfolio. We finally compare the percolation model, Gaussian and Student t4 copulas to market data and from our results determine that the percolation model provides a better fit to market data than the Gaussian and Student t4 copulas. Acknowledgements My utmost gratitude goes to my thesis supervisor, Prof. James Gleeson for allowing me to join his team, for his expertise, kindness, and most of all, for his patience. One simply could not wish for a better supervisor. I would like to express my thanks to Dr. Sergey Melnik for his helpful advice and comments during the past two years and a big thank you to Dr. Bernard Murphy from the Kemmy Business School for his help on the finance aspect of my thesis. I thank the academic and administrative staff of the Department of Mathematics and Statistics, UL for providing the environment to facilitate this work and especially Dr. Patrick Johnson for his LTEX expertise. I gratefully acknowledge funding provided by Science Foundation Ireland under programmes 06/IN.1/I366 and MACSI 06/MI/005. Lastly, I would like to thank my boyfriend Kieran and my family for their endless support, love and encouragement. For my parents who have supported and encouraged me to do my best in all matters of life, to them I dedicate this thesis.