A note on the Nelson-Siegel family

We study a problem posed in Bjork and Christensen (1999): Does there exist any nontrivial interest rate model that is consistent with the Nelson-Siege l family? They show that within the Heath-Jarrow-Morton framework with dete rministic volatility structure the answer is no. In this paper we give a generalized version of this result including stocha stic volatility structure. For that purpose we introduce the class of consi stent stare space processes, which have the property to provide an arbitrag e-free interest rate model when representing the parameters of the Nelson-S iegel family. We characterize the consistent state space Ito processes in t erms of their drift and diffusion coefficients. By solving an inverse probl em we find their explicit form. It turns out that there exists no nontrivia l interest rate model driven by a consistent state space Ito process.