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.

A Note on the Distribution of Multivariate Brownian Extrema

This paper presents a closed-form solution for the joint probability of the endpoints and minimums of a multidimensional Wiener process for some correlation matrices. This is the only explicit expressions found in the literature for this joint probability. The analysis can only be carried out for special correlation structures as it is related to the fundamentals regions of irreducible spherical simplexes generated by reflections and the link to the method of images. This joint distribution can be used in financial mathematics to obtain prices of credit or market related products in high dimension. The solution could be generalized to account for stochastic volatility and other stylized features of the financial markets.