The Nelson–Siegel–Svensson No-Arbitrage Yield Curve Model

Abstract

It is shown that the requirement to satisfy the no-arbitrage conditions specifies the Nelson–Siegel–Svensson model in the sense that it gives the coefficients of this model their obvious economic interpretation: the free coefficient should be a function of term to maturity, and the other coefficients should depend on the market state variables which, in turn, are selective values of stochastic processes at the time point at which the time structure is designed. It is shown that the model is a member of the family of affine yield models and is generated by a two-dimensional model of a short-term interest rate for the Nelson–Siegel model or a four-dimensional model of a short-term interest rate for the Nelson–Siegel–Svensson model. It is further shown that the yield curve of the European Central Bank (ECB) does not satisfy the no-arbitrage conditions. For this one must add one more term to the yield curve. As the state variables, it is necessary to choose a four-dimensional diffusion Gaussian process. A fifth factor is determined from the no-arbitrage conditions. The version of the modification of the yield curve proposed here differs from the earlier proposed modifications in order to ensure the absence of arbitrage opportunities. The probability properties of the yield interest rates that are generated by the Nelson-Siegel and Nelson-Siegel–Svensson models are considered. It is shown that the Nelson-Siegel model does not differ from the traditional two-factor model of affine yield, the volatility of which does not depend on the market state variables. Accordingly, the Nelson-Siegel-Svensson model does not differ from a four-factor model. These models generate the interest rates of yield to maturity and the forward yield with a normal distribution, for which the expectations and covariance matrices are found. To estimate the values of the rates of yield to maturity in the current time, a recurrent procedure is proposed based on the use of the Kalman filter.