"A Fast Algorithm for Computing Expected Loan Portfolio Tranche Loss in the Gaussian Factor Model"

 and

"Using Hermite Expansions for Fast and Arbitrarily Accurate Computation of the Expected Loss of a Loan Portfolio Tranche in the Gaussian Factor Model".

You can request the code using the form below.
Any updates to the code will be made available here.

The articles were reviewed by Mike Staunton in July 2006 issue of Wilmott magazine: 

ARTICLE 24: Back to Normal for CDOs

In March 2005, I set my students at CASS as assignment on the valuation of CDOs (collateralized debt obligations) for which I had no idea of the answer. Well for some weeks they put a lot of effort into tracking down the best methods they could find (such as the July 2004 paper by Michael Gibson of the Federal Reserve Board) before coding them up.

When they came to present their completed VBA functions, the most revealing aspect of their functions was the great difference in running time between the slowest (670 seconds) and the quickest (around 10 seconds) based on a reference portfolio containing 100 credits. Gibson’s model assumes that all the reference credits share the same correlation, default hazard rate and recovery rate. Then the unconditional default distribution is solved as an integral of the product of the conditional default distribution and a probability density function. Given the assumption of homogeneity for the reference credits, the expression for the conditional default distribution can be much simplified and this was the key element that the best code incorporated. It really pleases me when such a simplification of the maths produces such a great reduction in speed of calculation. As a result of this simplification, the finished code contains many uses of the COMBIN function and some might also use an approximation for this to speed up the code even further without losing accuracy. After that, one could try replacing the trapezium rule used for the integration with the more sophisticated Gaussian quadrature approach.

Some months later, I came across Pavel Okunev’s fast algorithm for computing the expected losses that lie at the heart of CDO valuation. A very similar approach had been given in internal Citigroup research by David Shelton dated August 2004. Both these approaches improve of the Gibson model by allowing the reference credits to have different asset correlations, default probabilities and recovery rates. The key assumption that both Okunev and Shelton make is that, conditional on the single shock factor, the joint distribution of losses is well approximated by a multivariate normal distribution. Here we see just another example of the central limit theorem, this time giving us the familiar normal approximation to the binomial (think Leisen and Reimer).

For smaller portfolios (say with between 25 and 50 credits), Okunev proposes that a Hermite expansion with up to 5 terms be used for more accurate results. It goes without saying that this normal approximation calculates in an instant and Shelton asserts that this proxy integration for valuing a CDO can be done in 1/25^th of a second on a desktop PC compared to some 220 seconds for Monte Carlo with 1,000,000 paths.