This paper studies the gains, in terms of served requests, attainable through out-of-band device-to-device (D2D) video exchanges in large cellular networks. A stochastic framework, in which users are clustered to exchange videos, is introduced, considering several aspects of this problem: the video-caching policy, user matching for exchanges, aspects regarding scheduling and transmissions. A family of \emph{admissible protocols} is introduced: in each protocol the users are clustered by means of a hard-core point process and, within the clusters, video exchanges take place. Two metrics, quantifying the "local" and "global" fraction of video requests served through D2D are defined, and relevant trade-off regions involving these metrics, as well as quality-of-service constraints, are identified. A simple communication strategy is proposed and analyzed, to obtain inner bounds to the trade-off regions, and draw conclusions on the performance attainable through D2D. To this end, an analysis of the time-varying interference that the nodes experience, and tight approximations of its Laplace transform are derived.