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 to show what is the impact of an improper calibration of the physical measure on the estimation of the empirical pricing kernel. While extracting the risk-neutral measure from option data provides a naturally forward looking measure, extracting the real world probability from a stream of historical returns is only partially informative, thus suboptimal with respect to investors’ future beliefs. In virtue of this disalignment, most of papers present in literature are then affected by a non-homogeneity bias. From a probabilistic viewpoints, the missing beliefs are totally unaccessible stopping times on the coarser filtration set. As a consequence, an absolutely continuous local or strict local martingale, once projected on it, becomes continuous with jumps. As a result of a non fully informative physical measure, the proposed empirical pricing kernel is no longer a true martingale, as required by the classical theory, but a strict local martingale with consequences on the probabilistic nature of the relative risk neutral measure. Finally we show how the implied options’ moments help in reducing the degree of inaccessibility and shorten the distance between what is theoretically required and empirically accessible.