We develop a GARCH option model with a new pricing kernel allowing for a variance premium. While the pricing kernel is monotonic in the stock return and in variance, its projection onto the stock return is nonmonotonic. A negative variance premium makes it U shaped. We present new semiparametric evidence to confirm this U-shaped relationship between the risk-neutral and physical probability densities. The new pricing kernel substantially improves our ability to reconcile the time-series properties of stock returns with the cross-section of option prices. It provides a unified explanation for the implied volatility puzzle, the overreaction of long-term options to changes in short-term variance, and the fat tails of the risk-neutral return distribution relative to the physical distribution.
Christoffersen, P., Heston, S., & Jacobs, K. (2013). Capturing Option Anomalies with a Variance-Dependent Pricing Kernel. Review of Financial Studies, 26(8), 1963-2006. https://doi.org/10.1093/rfs/hht033