Revisiting the fitting of the Nelson-Siegel and Svensson models

The Nelson-Siegel and the Svensson models are two of the most widely used models for the term structure of interest rates. Even though the models are quite simple and intuitive, fitting them to market data is numerically challenging and various difficulties have been reported. In this paper, a novel mathematical analysis of the fitting problem based on parametric optimisation is carried out. The analysis is based on the known observation that the fitting problem can be formulated as a separable nonlinear least-squares problem, in which the linear parameters can be eliminated. We specifically provide a thorough discussion on the conditioning of the inner part of the reformulated problem and show that many of the reported difficulties encountered when solving it are inherent to the problem formulation itself and cannot be tackled by choosing a particular optimisation algorithm.

Our stability analysis provides novel insights that we then use to show that some of the ill-conditioning of the problem can be avoided, and that a suitably chosen penalty approach can be used to take care of the remaining ill-conditioning.  As our numerical results indicate, this approach has indeed the expected impact, while being fully independent of any choice of a particular optimisation algorithm. We further establish smoothness and differentiability properties of the reduced objective function, which for the first time puts global optimisation methods for the reduced problem on a sound mathematical basis.



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