Calibration of a multifactor model for the forward markets of several commodities.

We propose a model for the evolution of forward prices of several commodities, which is an extension of the factor forward model in [3, 10], to a market where multiple commodities are traded. We calibrate this model in a market where forward contracts on multiple commodities are present, using historical forward prices. First we calibrate separately the four coefficients of every single commodity, using an approach based on quadratic variation/covariation of forward prices. Then, with the same technique, we pass to estimate the mutual correlation among the Brownian motions driving the different commodities. This calibration is compared to a calibration method used by practitioners, which uses rolling time series and requires a modification of the model, but turns out to be more accurate in practice, especially with a low frequency of observed transaction. We present efficient methods to perform the calibration with both methods, as well as the calibration of the intercommodity correlation matrix. Then we calibrate our model to WTI, ICE Brent and ICE Gasoil forward prices. Finally we present how to estimate spot volatility from forward parameters, with an application to the WTI spot volatility.