CONSISTENT PARALLEL AND PROPORTIONAL SHIFTS IN THE TERM STRUCTURE OF FUTURES PRICES

We consider an arbitrage-free futures price model of Heath–Jarrow–Morton type which is driven by a multidimensional Wiener process and a marked point process. We find necessary and sufficient conditions for this model to produce a log futures curve that changes only through parallel shifts. The same analysis is carried out for the case when the log futures curve changes only through proportional shifts. We prove that there exist nontrivial parallel and proportional shifting log futures curves and we show how to specify the futures price model in order to obtain them. Additionally the shift functions are characterized. Finally, we consider the case of all other single-factor affine models which are neither parallel nor proportional shifting curves. We find necessary and sufficient conditions for the purely Wiener-driven log futures model to admit such other affine shifting curve and we characterize the shift functions.