On finite dimensional realizations for the term structure of futures prices

TLDR: In this paper, the authors consider HJM type models for the term structure of futures prices, where the volatility is allowed to be an arbitrary smooth functional of the present futures price curve and provide necessary and sufficient conditions for when the induced spot price is a Markov process.

Abstract: We consider HJM type models for the term structure of futures prices, where the volatility is allowed to be an arbitrary smooth functional of the present futures price curve. Using a Lie algebraic approach we investigate when the infinite dimensional futures price process can be realized by a finite dimensional Markovian state space model, and we give general necessary and sufficient conditions, in terms of the volatility structure, for the existence of a finite dimensional realization. We study a number of concrete applications including a recently developed model for gas futures. In particular we provide necessary and sufficient conditions for when the induced spot price is a Markov process. In particular we can prove that the only HJM type futures price models with spot price dependent volatility structures which generically possess a spot price realization are the affine ones. These models are thus the only generic spot price models from a futures price term structure point of view.