We introduce a new class of market models for underlying assets given by a pure jump martingale. We capture the information contained in the surface of call options in the density of the Levy measure of an additive process, and we set this static code in motion by means of a stochastic dynamics of Ito’s type in a function space, creating what we call a ”tangent Levy model”. We characterize the consistency of these models by a drift condition ”a la HJM”, and we construct explicit examples of consistent tangent Levy models.