Pricing of Options on Commodity Futures with Stochastic Term Structures of Convenience Yields and Interest Rates

We develop a model to value options on commodity futures in the presence of stochastic interest rates as well as stochastic convenience yields. In the development of the model, we distinguish between forward and future convenience yields, a distinction that has not been recognized in the literature. Assuming normality of continuously compounded forward interest rates and convenience yields and log-normality of the spot price of the underlying commodity, we obtain closed-form solutions generalizing the Black-Scholes/Merton's formulas. We provide numerical examples with realistic parameter values showing that both the effect of introducing stochastic convenience yields into the model and the effect of having a short time lag between the maturity of a European call option and the underlying futures contract have significant impact on the option prices.