This paper introduces a two-factor continuous-time model for commodity pricing under the assumption that prices revert to a stochastic mean level, which shows smooth, periodic fluctuations over long periods of time. We represent the mean reversion price by a Fourier series with a stochastic component. We also consider a seasonal component in the price level, an essential characteristic of many commodity prices, which we represent again by a Fourier series. We obtain analytical pricing expressions for futures contracts. Using futures price data on Natural Gas, we provide evidence on the presence of long-term fluctuations and show how to estimate the long-term component simultaneously with a seasonal component using the Kalman filter. We analyse the in-sample and out-of-sample empirical performance of our pricing model with and without a seasonal component and compare it with Schwartz and Smith (2000) model. Our findings show the in-sample and out-of-sample superiority of our model with seasonal fluctuations, thereby providing a simple and powerful tool for portfolio management, risk management, and derivative pricing.

Introduction

Characterising the stochastic behaviour of commodity prices is an issue of special relevance for practitioners in financial markets, since some commodity markets are very liquid and trade high volumes every day. Markets for futures, options, and options on futures with some commodity as the underlying asset are also very active. Commodities are not standard financial assets, so it should not be surprising that they might need specific valuation models. In particular, the characteristics of many commodity products and markets imply a relatively complex pricing nature that may combine short-term seasonal behaviour with fluctuations around a long-term trend. Short-term seasonal fluctuations over an annual period generally reflect changes in demand and supply across the different seasons in a year (Gould et al., 2008; Taylor, 2010). However, changes in production technology or shifts in taste may give rise to long-term trends that cause market prices to fluctuate. Indeed, in line with our argument, Mu and Ye (2015) analysed the crude oil market and found evidence of a long-term trend combined with cyclical movements. The goal of this paper is to propose a model for commodity prices that can help characterise and estimate such components when they are present, without imposing any a priori constraint on their periodicity.