On the Volatility of Commodity Futures Prices

Abstract

This paper analyzes the volatility structure of commodity futures markets by using a continuous time forward price model with stochastic volatility. The model features three distinct volatility structures, each one potentially assessing the impact of long-term, medium-term and short-term variation, respectively. Using an extensive 21 year database of commodity futures prices, the model is estimated for six key commodities: gold, crude oil, natural gas, soybean, sugar and corn. The model is well suited to identify the shape and the persistence of each volatility factor, their contribution to the total variance, the extent to which commodity futures volatility can be spanned, and the nature of the return-volatility relation.

Appendix: Proof of Theorem 1

We consider the process $X(t,T) = \ln F(t,T, \mathbf {V_t})$, where the forward price dynamics are given by (1) with the volatility specifications (2). Then an application of the Ito’s formula derives

$$\begin{aligned} F(t,T, \mathbf {V_t}) = F(0,T) \exp \left[ \sum _{i=1}^3 \int \limits _0^t \sigma _i(s,T, \mathbf {V_s}) dW_i(s) -\frac{1}{2} \sum _{i=1}^3 \int \limits _0^t \sigma ^2_i(s,T,\mathbf {V_s} ) ds \right] . \end{aligned}$$

(14)

By introducing the state variables

$$\begin{aligned} \phi _1(t)&= \int \limits _0^t \sqrt{\mathbf {V^1_s}} dW_1(s), \qquad \mathsf x _1(t) = \int \limits _0^t \mathbf {V^1_s} ds \\ \phi _2(t)&= \int \limits _0^t e^{-\eta _2 (t-s)} \sqrt{\mathbf {V^2_s}} dW_2(s), \qquad \mathsf x _2(t) = \int \limits _0^t e^{-2 \eta _2 (t-s)} \mathbf {V^2_s} ds \\ \phi _3(t)&= \int \limits _0^t e^{-\eta _3(t-s)}\sqrt{\mathbf {V^3_s}}dW_3(s), \qquad \phi _4(t) = \int \limits _0^t (t-s)e^{-\eta _3(t-s)}\sqrt{\mathbf {V^3_s}}dW_3(s) \\ \mathsf x _3(t)&= \int \limits _0^t e^{-2\eta _3(t-s)} \mathbf {V^3_s} ds, \qquad \mathsf x _4(t) = \int \limits _0^t (t-s)e^{-2\eta _3(t-s)} \mathbf {V^3_s} ds, \\ \mathsf x _5(t)&= \int \limits _0^t (t-s)^2e^{-2\eta _3(t-s)} \mathbf {V^3_s} ds. \end{aligned}$$

and performing some basic manipulations, Eq. (14) can be expressed as (7).