Optimization problems with discrete decisions are nonconvex and thus lack strong duality, which limits the usefulness of tools such as shadow prices. It was shown in Burer (2009) that mixed-binary quadratic programs can be written as completely positive programs, which are convex. We apply this perspective by writing unit commitment in power systems as a completely positive program, and then using the dual copositive program and strong duality to design new pricing mechanisms. We show that the mechanisms are revenue-adequate, and, under certain conditions, support a market equilibrium. To facilitate implementation, we also employ a cutting plane algorithm for solving copositive programs exactly, which we further speed up via a second-order cone programming approximation. We provide numerical examples to illustrate the potential benefits of the pricing mechanisms and algorithms.