Long-term Futures Curves and Seasonal Structures of Wheat in the European Union and the United States

A two-factor affine theoretical model is used to estimate the long-term futures curves for wheat in the European Union and the United States, as represented by the Euronext and CME markets, respectively. The CME futures curve exhibits a long-term equilibrium; in contrast, the Euronext futures curve does not show a tendency for futures to revert to a long-term equilibrium value. The estimated seasonality is relatively similar for both markets. However, the seasonal minimum and maximum points in the futures curve occur one to two months later for Euronext compared to the CME. More importantly, the futures curve for Euronext has a much more marked seasonality than the CME futures curve. Credible intervals of the futures curves are also estimated. The width clearly increases for longer maturities, but it does so much faster for Euronext than for the CME. For long-maturity futures, variability in the parameter estimates (as opposed to the residual errors) accounts for most of the width of the credible intervals, especially for Euronext. The proposed model can be used to price long-term futures options, long-term price insurance, and long-term swaps, among other applications.