Parametric Yield Curve Models: Splines, Nelson-Siegel, and Svensson

The bootstrapped curve is exact — it prices every input instrument perfectly — but the implied forward curve is jagged and unusable for option pricing or monetary policy analysis. A smooth forward curve requires imposing additional structure. The two dominant approaches are:

1. Non-parametric smoothing — cubic B-splines fitted to the discount function, with regularization that penalizes curvature.
2. Parametric families — Nelson-Siegel and Svensson, which model the forward curve as a sum of exponential-polynomial basis functions with 4–6 parameters.

The Nelson-Siegel family is used by the Federal Reserve, the ECB, the Bank of England, and most central banks worldwide for publishing yield curve estimates. Understanding it is prerequisite to interpreting the level, slope, and curvature factors that practitioners and policymakers read off the curve daily.

• state the $p = Cd + \varepsilon$ estimation problem and explain the role of the pricing error vector;
• describe how cubic B-splines parameterise the discount function and why they produce smooth forward curves;
• write down the Nelson-Siegel forward curve $\phi_{NS}(x; z)$ and integrate it to get the spot rate;
• interpret the four parameters of Nelson-Siegel as level, slope, hump height, and decay;
• state the Svensson extension and explain why it offers more flexibility;
• list the four desiderata for a good curve family (flexible, parsimonious, regular, consistent);
• name two central banks that use each family.

1. 1. From exact-fit to smooth-fit
2. 1. Cubic B-splines
3. 1. Nelson-Siegel family
4. 1. Svensson extension
5. 1. Desiderata for a curve family
6. 1. Fitting in practice