The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging

Abstract

In this article we compare three models of the stochastic behavior of commodity prices that take into account mean reversion, in terms of their ability to price existing futures contracts, and their implication with respect to the valuation of other financial and real assets. The first model is a simple one‐factor model in which the logarithm of the spot price of the commodity is assumed to follow a mean reverting process. The second model takes into account a second stochastic factor, the convenience yield of the commodity, which is assumed to follow a mean reverting process. Finally, the third model also includes stochastic interest rates. The Kalman filter methodology is used to estimate the parameters of the three models for two commercial commodities, copper and oil, and one precious metal, gold. The analysis reveals strong mean reversion in the commercial commodity prices. Using the estimated parameters, we analyze the implications of the models for the term structure of futures prices and volatilities beyond the observed contracts, and for hedging contracts for future delivery. Finally, we analyze the implications of the models for capital budgeting decisions.

A Two-Factor Model for Commodity Prices and Futures Valuation

“…This assumption is standard and is also followed by Schwartz (1997). The process for the log price then becomes:…”

“…The convenience yield is brought into the spot price process as a dividend yield. Schwartz (1997) further explores this model empirically by adopting a more sophisticated calibration method and tests it with several commodities. Schwartz (1997) three-factor model, Miltersen and Schwartz (1998) and Hilliard and Reis (1998) add a third stochastic factor to the model to account for stochastic interest rates.…”

“…Due to the non-observability of the state variables, the linearity of the logarithm of the futures prices in the state variables and the Markovian property of these, the Kalman filter seems to be the most appropriate technique to estimate the model parameters. This method is also applied by Schwartz (1997). The basic principle of Kalman filter is the use of temporal series of observable variables to reconstitute the value of the non-observable variables.…”

“…Additionally, we also apply the Kalman filter to the Gibson and Schwartz (1990) - Schwartz (1997) two factor model using the same data set for estimation purposes and to compare the results. We also compare the futures volatility structure implied by the data and by both our model and Schwartz model.…”

“…In fact, our model closely resembles the Heston (1993) stochastic volatility model for standard asset prices and therefore can be interpreted as the storable commodity version of this model 3 . The second difference lies in the nature of the data used to test the models empirically, which follows Gibson and Schwartz (1990) - Schwartz (1997). Nielsen and Schwartz (2004) use copper futures data to calibrate their model while we use light crude oil futures data.…”