Optimal derivatives design for mean-variance agents under adverse selection

We consider a problem of derivatives design under asymmetry of information: the principal sells a contingent claim to an agent, the type of whom he does not know. More precisely, the principal designs a contingent claim and prices it for each possible agent type, in such a way that each agent picks the contingent claim and pays the price that the principal designed for him. We assume that the preferences of the agent depend linearly on the parameters which determine the agent's type; this model is rich enough to accommodate quadratic utilities. The problem then is reformulated as an optimization problem, where the optimization is performed within a class of convex functions. We prove an existence result for the provide explicit examples in the case when the agent is fully characterized by a single parameter.