Commodity Modelling*

From the supply-and-demand relation, the time-t commodity futures price \(F_t\) for a contract with expiration date T is \(\displaystyle S_te^{(r-y)(T-t)}, \) where y is a number reflecting both convenience yield and storage costs, so it can be positive or negative. For simplicity, we still call it convenience yield in modelling. One way for commodity modelling is to assume that the spot price follows a geometric Brownian motion under the risk-neutral measure \(\displaystyle dS_t=(r-y)S_t dt+\sigma S_t d\widetilde {W}_t, \) which leads to \(\displaystyle dF_t=\sigma F_td\widetilde {W}_t, \) showing that the futures price \(F_t\) is a martingale. It is used for European options that depend mainly on final distributions. However, for path-dependent or some other exotic options, more advanced models may be required to capture the mean-reverting property of some of the commodities as well as the volatility skew.