Technical Note: Simulating the Two-Factor Schwartz and Smith Model of Commodity Prices

Abstract

Commodity price simulation is useful in many engineering economics applications, yet discrete approximations of the continuous stochastic processes used in modeling commodity prices are not always straightforward. This article describes the exact solution for discretely simulating the Schwartz and Smith (2000) two-factor model of commodity prices.

Acknowledgments

I would like to thank three anonymous reviewers for suggestions that have improved the article. I also thank William Navidi for assistance with the proof in footnote 3 and Jim Smith for his assistance in understanding the nuances associated with simulating the Schwartz and Smith model.

Notes

The Schwartz and Smith model has been shown to be equivalent to the Gibson and Schwartz (1990) stochastic convenience yield model (Schwartz and Smith 2000). Gibson and Schwartz (1990) is the foundation for the two-factor model in Schwartz (1997). Other versions of this two-factor commodity price model include Baker et al. (1998), Pilipovic (1998), and Pindyck (1999). We review the Schwartz and Smith model because it is the most popular nonstationary two-factor model currently in use in commodity price simulations.

The author's name has been withheld because the point here is simply to note an example of the mistakes being made. The Anon paper was published in a respected refereed economics journal.

Two random draws ϵ∼N(0, 1) and ϖ∼N(0, 1) that are to be correlated with correlation coefficient ρ can be generated via taking random draws ϵ∼N(0, 1) and ϖ∼N(ρϵ, 1−ρ²). The proof is as follows: Given two independent standard normal random variables z ₁ and z ₂, define ϵ and ϖ by ϵ=z ₁ and . This ensures that ϖ∼N(ρϵ, 1−ρ²). Now ϵ∼N(0, 1), and because and Var(ω)=ρ² Var(ϵ)+(1−ρ²)Var(z ₂)=1, ϖ∼N(0, 1) as well. Because ϵ and ϖ have mean 0 and variance 1, the correlation between ϵ and ϖ is . Because z ₁ and z ₂ are independent, and because ϵ=z ₁, E(ϵz ₂)=E(ϵ)E(z ₂)=0. Because E(ϵ²)=Var(ϵ)=1, E(ϵω)=ρ.

Once the optimal asset management is decided and the consequent value of the project is calculated, the true processes are used to compute event probabilities and for other planning purposes. For a nontechnical review of dynamic discounted cash flow analysis and contingent claims analysis, see Samis et al. (2006).

For example, the 10th percentile would be .

Note that Figure 1 in SS lists the slope of the long-run equilibrium price being . It should say that the rate of growth of the long-run equilibrium price is .

For example, the 10th percentile would be .

See Schwartz and Smith (2000) and Lucia and Schwartz (2002) for methods of estimating these parameters for a particular price time series. Lo and Wang (1995) and Campbell et al. (1997) provide comprehensive details on estimating two-factor systems.