Efficient wavelets-based valuation of synthetic CDO tranches

We present new formulae for the valuation of synthetic collateralized debt obligation (CDO) tranches under the one-factor Gaussian copula model. These formulae are based on the wavelet theory and the method used is called WA. We approximate the cumulative distribution function (CDF) of the underlying pool by a finite combination of jth order B-spline basis, where the B-spline basis of order zero is typically called a Haar basis. We provide an error analysis and we show that for this type of distributions, the accuracy in the approximation is the same regardless of the order of the B-spline basis employed. The resulting formula for the Haar basis case is much easier to implement and performs better than the formula for the B-spline basis of order one in terms of computational time. The numerical experiments confirm the impressive speed and accuracy of the WA method equipped with a Haar basis, independently of the inhomogeneity features of the underlying pool. The method appears to be particularly fast for multiple tranche valuation.