The laws of two continuous martingales will typically be singular to each other and hence have infinite relative entropy. But this does not need to happen in discrete time. This suggests defining a new object, the specific relative entropy, as a scaled limit of the relative entropy between the discretized laws of the martingales. This definition goes all the way back to Gantert's PhD thesis, and in recent time Follmer has rekindled the study of this object by for instance obtaining a novel transport-information inequality.

In this talk I will first discuss recent results on explicit formulae (or bounds) for the specific relative entropy in terms of the quadratic variation processes of the martingales involved. Next I will describe an application of this object to prediction markets. Concretely, D. Aldous asked in an open question to determine the 'most exciting game', i.e. the prediction market with the highest entropy. We formalize this question as a problem of specific relative entropy optimization and completely characterize its optimizer. As a crucial step, we make an unexpected connection to the field of Monge-Ampère equations.