Co-monotonicity of optimal investments and the design of structured financial products

We prove that, under very weak conditions, optimal financial products on complete markets are co-monotone with the reversed state price density. Optimality is meant in the sense of the maximization of an arbitrary preference model, e.g., expected utility theory or prospect theory. The proof is based on a result from transport theory. We apply the general result to specific situations, in particular the case of a market described by the Capital Asset Pricing Model or the Black–Scholes model, where we derive a generalization of the two-fund-separation theorem and give an extension to APT factor models and structured products with several underlyings. We use our results to derive a new approach to optimization in wealth management, based on a direct optimization of the return distribution of the portfolio. In particular, we show that optimal products can (essentially) be written as monotonic functions of the market return. We provide existence and nonexistence results for optimal products in this framework. Finally we apply our results to the study of bonus certificates, show that they are not optimal, and construct a cheaper product yielding the same return distribution.