A closed-form approximation for pricing spread options on futures under a mean-reverting spot price model with multiscale stochastic volatility

Commodity spot prices tend to revert to some long-term mean level and most commodity derivatives are based on futures prices, not on spot prices. So, we consider spread options on futures instead of spot or spot index, where the log spot price follows a mean-reverting process. The volatility of the mean-reverting process is driven by two different (fast and slow) scale factors. We use asymptotic analysis to obtain a closed-form approximation of the futures prices and a closed-form formula for the approximate prices of spread options on the futures. The overall improvement of our analytic formula over the classical Kirk-Bjerksund-Sternsland (KBS) formula is discussed via numerical experiments.