Efficient wavelets-based valuation of synthetic CDO tranches

Abstract

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$ ^{[a,b]}$ . We approximate the cumulative distribution function (CDF) of the underlying pool by a finite combination of $ j$ th 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$ ^{[a,b]}$ 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.