Are We Extracting the True Risk Neutral Density from Option Prices? A Question with No Easy Answer

In this paper we raise a question on the theoretical foundation of option implied risk neutral density. We prove that given any number of options, there exist numerous risk neutral densities which are piecewise constant, have only two values, either an lower bound or an upper bound on the true risk neutral density, and price all these options correctly. We also prove that given any number of options, there exist numerous risk neutral densities consistent with the prices of all these options whose first derivatives are piecewise constant and have only two values, either an lower bound or an upper bound on the true risk neutral density's first derivative. Similar results are proved with respect to the true risk neutral density's higher order derivatives. These results show how large errors we can make when extracting RNDs from option prices