Optimal contract design via relaxation: application to the problem of brokerage fee for a client with private signal

Abstract In this paper, we show how relaxation techniques can be used to establish the existence of an optimal contract in the presence of information asymmetry. The method we illustrate was initially motivated by the problem of designing optimal brokerage fees, but it does apply to other optimal contract problems in which (i) the agent controls linearly the drift of a diffusion process, (ii) the direct dependence of the principal’s and the agent’s objectives on the strategy of the agent is of a special form, and (iii) the space of admissible contracts is compact. This method is then applied to establish the existence of an optimal brokerage fee in a market model with a private trading signal observed by the broker’s client, but not by the broker.