We study a problem posed in Bj"ork and Christensen (1999): Does there exist any nontrivial interest rate model that is consistent with the Nelson–Siegel family? They show that within the Heath–Jarrow–Morton framework with deterministic volatility structure the answer is no.
In this paper we give a generalized version of this result including stochastic volatility structure. For that purpose we introduce the class of consistent state space processes, which have the property to provide an arbitrage-free interest rate model when representing the parameters of the Nelson–Siegel family. We characterize the consistent state space Itô processes in terms of their drift and diffusion coefficients. By solving an inverse problem we find their explicit form. It turns out that there exists no nontrivial interest rate model driven by a consistent state space Itô process.