Sentiment lost : the effect of projecting the empirical pricing kernel onto a smaller filtration set

Supported by empirical examples, this paper provides a theoretical analysis on the impacts of using a suboptimal information set for the estimation of the empirical pricing kernel and, more in general, for the validity of the fundamental theorems of asset pricing. While inferring the risk-neutral measure from options data provides a naturally forward- looking estimate, extracting the real-world one from a stream of historical returns is only partially informative, thus suboptimal with respect to investors' future beliefs. As a consequence of this disalignment, the two measures no longer share the same nullsets thus distorting the investor's risk premium. It follows that the relative empirical pricing kernel is no longer a true martingale, as required by the theory, but a strict local martingale with consequences on the validity of the risk-neutral pricing. From a probabilistic viewpoints, the missing beliefs are totally unaccessible stopping times on the coarser filtration set so that an absolutely continuous strict local martingale, once projected on it, becomes continuous with jumps