On the comparison of Schwartz and Smith’s two- and three-factor models on commodity prices

Since Schwartz and Smith (2000) published their study on two-factor model on commodity prices, many studies have used this model and others have extended it. The authors also proposed the three-factor model due to the poor fitting of the two-factor one on long-term futures prices. At that time the authors had only long-term prices from a private source to calibrate, test and compare these models. No public data on long-term future contracts were available. On the other hand, during the last decade the commodity prices soared as did the liquidity of long-term contracts. This means that the interest of the agents in the management of their risk on long-term positions increased the same way and this is the motivation for this study. In this article, we revisit the comparison between two- and three-factor models using public data for short- and long-term contracts (we use up to the 67-month-ahead contract). We also provide a detailed derivation of the three-factor model differently from that of the original article. Following the original article of Schwartz and Smith, we used oil futures prices traded on the New York Mercantile Exchange to calibrate the model. The results show a better fit of the three-factor model for the term structure of prices and volatilities mainly for long maturities contracts, while the two-factor model in most portions of the curve underestimates the risk premiums. This type of analysis is important not only for daily agents negotiating the physical commodities through long-term contracts but also for investment decisions on development of real projects.