Customizable Pecuniary Risk and Market Incompleteness Premium

We examine the optimal customization of a financial derivative in the presence of a background risk. This problem includes the model of finding the optimal constant amount of a given pecuniary risk as a degenerated case. We show the importance of this perspective with a preference-free solution for the pecuniary background risk case and a general solution for any given utility function. The latter solution allows us to measure the incompleteness premium demanded by risk-averse agent for a market where only constant amount of risk is allowed in comparison with a complete and arbitrage-free market where the customization is unrestricted.