Energy commodities and calendar spread options

We present a unified framework for pricing calendar spread options on energy commodities under affine models featuring stochastic volatility, jumps, and Samuelson effect. Expressions for the joint characteristic function of log-futures prices are derived, enabling efficient calibration and valuation. An empirical analysis, across WTI crude oil, HH natural gas, and ULSD heating oil shows that stochastic volatility models consistently outperform others. Jumps enhance short-term fit, while volatility dynamics matter more at longer maturities. The Black model remains competitive for short- and mid-term contracts.