Normal approximation for locally dependent collateralized debt obligation

Abstract

This dissertation sheds light on a collateralized debt obligation (CDO) since it resulted in the financial crisis between 2007 and 2008. Because of damage of this crisis, several authors have attempted to approximate loss on a tranche of a CDO. For instance, in 2009, Karoui and Jiao used a normal random variable to approximate the loss on a tranche of a CDO containing independent assets. However, in this work, we are attentive to dependence structure among assets in a CDO. We present two types of dependent condition: local dependence (LD) and disjoint local dependence (DLD). They roughly mean that defaults of some assets may influence defaults of other assets in their neighborhood but some assets are not correlated. An average loss on a tranche of a CDO is approximated by an average of a call function for the standard normal random variable. The uniform and non–uniform bounds are presented under the LD and DLD conditions by using the Stein’s method. Moreover, we illustrate two examples under the DLD condition and propose numerical bounds.