This paper sheds new light on higher-order price risks in crude oil markets. A model-free analysis reveals that crude oil variance risk behaves fundamentally different from variance risk in equity markets. Most importantly, a skewness swap is no valid hedge for a variance swap and investors fear large price jumps in both directions. A model-based assessment confirms this and reveals that while stochastic volatility is important to capture the statistical properties such as volatility clusters and time-varying variance swap rates, only jump risk seems to be priced with a premium. Empirical evidence from a pricing and hedging exercise confirms these findings.