NelsonSiegelSvensson

The Nelson-Siegel-Svensson model, often abbreviated NSS, is a parsimonious parametric representation used to describe the term structure of interest rates. It provides a compact formula to estimate the yield for any given maturity by fitting a small number of parameters, enabling smooth interpolation and extrapolation of the yield curve. The model extends the earlier Nelson-Siegel form by incorporating an additional curvature term, an extension introduced by Lars Svensson to improve fit across a broader range of maturities.

Let y(tau) denote the yield for maturity tau. The NSS specification is

y(tau) = beta0

+ beta1 * [(1 - exp(-tau/lambda1)) / (tau/lambda1)]

+ beta2 * [(1 - exp(-tau/lambda1)) / (tau/lambda1) - exp(-tau/lambda1)]

+ beta3 * [(1 - exp(-tau/lambda2)) / (tau/lambda2) - exp(-tau/lambda2)],

where lambda1 > 0 and lambda2 > 0 are decay parameters, and beta0, beta1, beta2, beta3 are shape parameters. The first three terms form the Nelson-Siegel component (level, slope, and curvature), while the beta3 term adds an extra curvature feature through lambda2.

In practice, NSS parameters are estimated from observed yields across maturities, typically by nonlinear least squares or time-series methods that allow for evolving curves. The model is valued for its balance of flexibility and parsimony, making it a common tool for central banks, pension funds, and other financial institutions to construct discount curves, price interest rate derivatives, and forecast forward rates. Limitations include potential instability outside calibration ranges and sensitivity to the choice of lambda parameters.