Bilevel Portfolio Optimization with Ordered Pricing Models

Pricing problems consist in determining the optimal prices to charge some services taking into account the reaction of customers to prices, as they could modify their purchases to avoid costs. Here we consider a pricing problem in the financial industry, namely, between the broker and the investor, as the former fixes the transaction costs on assets that are purchased by the latter. We assume that the broker defines a pricing policy based on ordering its fees and, depending on their amount, fees are fully or partially due by the investor. As a result, the function describing the total transaction costs is the ordered median and, to maximize its profit, the broker must determine the optimal weights of that function. In the model formulation, the broker must take into account investor's reaction, resulting in a bilevel optimization model. In this model, the first level is broker's optimal choice that is followed in the second level by investor's optimal choice. We show how to formulate and solve the bilevel problem assuming two different sets of possible prices: discrete and continuous. In the former case, the non-linear formulation is reduced into a mixed integer linear model, while in the latter case, the resulting model remains as a mixed integer non-linear model. The obtained solution let us suggest some managerial insights of pricing policies, as broker's profits and investor's reaction to prices can be predicted and described. We will see that there is a trade-off between broker's profits and investor's risk, as pricing policies favorable to the broker increases the financial risks of the investor. This suggest the broker to be cautious on implementing these policies, if investor's gains and losses are of concern to the broker