Bilateral Multiple Gamma Returns: Their Risks and Rewards

The bilateral gamma model for returns is naturally derived from the lognormal model. Maximizing entropy in a random time change delivers the symmetric variance gamma model. The asymmetric variance gamma follows on incorporating skewness. Differential speeds for the upward and downward motions lead to the bilateral gamma. A further generalizations results in the bilateral double gamma model when the speed parameter of the bilateral gamma model is itself taken to be gamma distributed on entropy maximization. A rich five to seven parameter specification of preferences renders possible the extraction of physical densities from option prices. The quality of such extraction is measured by examining the uniformity of the estimated distribution functions evaluated at realized forward returns. The economic value of risky returns is seen to embed three/five risk premia for the bilateral gamma/bilateral double gamma. For the bilateral gamma they are up and down side volatilities compensated in up and down side drifts, and the down side drift compensated in the up side drift. For the bilateral double gamma one adds in addition compensations for skewness. Results reveal a drop in the down side risk premium since the crisis with an increase in the recent period.