Numerical Methods for the Valuation of Synthetic

Numerical Methods for the Valuation of Synthetic Collateralized Debt Obligations Xiaofang Ma Doctor of Philosophy Graduate Department of Computer Science University of Toronto 2007 A Collateralized Debt Obligation (CDO) is a credit derivative that creates fixed income securities, which are known as tranches. A CDO is called a synthetic CDO if the risky assets in the underlying pool are credit default swaps. An essential part of the valuation of a synthetic CDO tranche is how to estimate accurately and efficiently the expected value of the tranche loss function. It is known that the expected value of a function of one random variable is completely determined by the distribution of the random variable and the function itself. A standard approach to estimate the expected value of a function of one random variable is to estimate the distribution of the underlying random variable, the pool loss in our case, and then to evaluate the expected value of the given function, the tranche loss function for our problem. Following this approach, we introduce three methods for estimating the distribution of the pool loss: a stable recursive method for computing the distribution of the pool loss exactly, an improved compound Poisson approximation method and a normal power approximation method for approximating the distribution of the pool loss. We also develop a new method that focuses on the tranche loss function directly. The tranche loss function is expressed simply in terms of two bases functions. Each of the two bases functions is a transformation of the hockey stick function h(x), where h(x) = 1− x if 0 ≤ x < 1 and 0 if x ≥ 1. By approximating the hockey stick function by a sum of exponentials, the tranche loss function is approximated by a sum of exponentials.