The dynamics of the higher moments of the index return distribution are critically important for modeling time-series and cross-sectional patterns in option prices. We demonstrate how multifactor models improve on simpler models by more adequately capturing the levels and dynamics of higher moments and allowing for richer dynam- ics in the levels and term structures of the risk premiums associated with the first four moments. Including a second and especially a third stochastic volatility factor greatly improves option fit, and the resulting skewness and kurtosis better match non- parametric benchmarks. Return jumps provide modest improvements in option fit and a higher equity risk premium, but their impact on higher moment risk premiums is small. All models we investigate struggle to match the term structure of risk-neutral skewness and kurtosis.