Putting Smiles Back to the Futures

Abstract

This article reports a practical approach to extend the classical Gabillon model to allow explicit modelling of commodity futures smiles. The original Gabillon model is first extended with a deterministic shift to fit the term structure of futures prices. The smile information of individual futures is extracted from the futures options markets in terms of the implied marginal distributions. An algorithm based on the copula technique is then developed to reconstruct the joint distribution of the underlying futures prices that is consistent with both the term structure of volatilities and the marginal distributions of individual futures. Examples are provided to illustrate the calibration procedure and options pricing.