[This paper shows that economic linkages among commodities create a source of long-term correlation between futures returns. We extend the theory of storage to a multi-commodity level and find that the convenience yield of a commodity depends on its relative scarcity with respect to other related commodities. This implies a feedback effect between commodities that is necessary to replicate the upward-sloping correlation term structure of futures returns observed for related commodities. We present a multi-commodity affine model that validates our theoretical predictions and considerably reduces the pricing errors in out-of-sample crack spread options.]

We characterize a three‐factor model of commodity spot prices, convenience yields, and interest rates, which nests many existing specifications. The model allows convenience yields to depend on spot prices and interest rates. It also allows for time‐varying risk premia. Both may induce mean reversion in spot prices, albeit with very different economic implications. Empirical results show strong evidence for spot‐price level dependence in convenience yields for crude oil and copper, which implies mean reversion in prices under the risk‐neutral measure. Silver, gold, and copper exhibit time variation in risk premia that implies mean reversion of prices under the physical measure.

We propose a parsimonious ‘unspanned stochastic volatility’ model of the term structure and study its implications for fixed-income option prices. The drift and quadratic variation of the short rate are affine in three state variables (the short rate, its long-term mean and variance) which follow a joint Markov (vector) process. Yet, bond prices are exponential affine functions of only two state variables, independent of the current interest rate volatility level. Because this result holds for an arbitrary volatility process, such a process can be calibrated to match fixed income derivative prices. Furthermore, this model can be ‘extended’ (by relaxing the time-homogeneity) to fit any arbitrary term structure. In its ‘HJM’ form, this model nests the analogous stochastic equity volatility model of Heston (1993) [Heston, S.L., 1993. A closed form solution for options with stochastic volatility. Review of Financial Studies 6, 327–343]. In particular, if the volatility process is specified to be affine, closed-form solutions for interest rate options obtain. We propose an efficient algorithm to compute these prices. An application using data on caps and floors shows that the model can capture very well the implied Black spot volatility surface, while simultaneously fitting the observed term structure.

We model oil price dynamics in a general equilibrium production economy with two goods: a consumption good and oil. Production of the consumption good requires drawing from oil reserves at a fixed rate. Investment necessary to replenish oil reserves is costly and irreversible. We solve for the optimal consumption, production and oil reserves policy for a representative agent. We analyze the equilibrium price of oil, as well as the term structure of oil futures prices. Because investment in oil reserves is irreversible and costly, the optimal investment in new oil reserves is periodic and lumpy. Investment occurs when the crude oil is relatively scarce in the economy. This generates an equilibrium oil price process that has distinct behavior across two regions (characterized by the abundance/scarcity of oil). We undertake three empirical tests suggested by our model. First, we estimate key parameters using SMM to match moments of oil price futures as well as other macroeconomic properties of the data. Second, we estimate an affine regime switching model of the oil price, which captures the main features of our equilibrium model and preserves the tractability of reduced-form models. Lastly, we compare the risk premium in short-maturity oil futures implied by our model to the data.

This paper shows that economic linkages among commodities create a source of long-term correlation between futures returns. We extend the theory of storage to a multi-commodity level and find that the convenience yield of a commodity depends on its relative scarcity with respect to other related commodities. This implies a feedback effect between commodities that is necessary to replicate the upward-sloping correlation term structure of futures returns observed for related commodities. We present a multi-commodity affine model that validates our theoretical predictions and considerably reduces the pricing errors in out-of-sample crack spread options.