Statistically consistent term structures have affine geometry

This paper is concerned with finite dimensional models for the entire term structure for energy futures. As soon as a finite dimensional set of possible yield curves is chosen, one likes to estimate the dynamic behaviour of the yield curve evolution from data. The estimated model should be free of arbitrage which is known to result in some drift condition. If the yield curve evolution is modelled by a diffusion, then this leaves the diffusion coefficient open for estimation. From a practical perspective, this requires that the chosen set of possible yield curves is compatible with any obtained diffusion coefficient. In this paper, we show that this compatibility enforces an affine geometry of the set of possible yield curves.