Abstract
Several simple multi-factor s.d.e. models have been proposed in the literature to capture different features of interest to particular commodity markets, with the key attention paid to the relationship between the spot price and futures curves under such models. Though these models are perhaps too stylized and simplistic for modelling applications on a wide range of commodity markets, they still pose a strong degree of interest as they can be extended to larger classes of model structures that make them practically relevant. However, until recently such approaches have not been widely adopted, largely due to the challenge involved in performing calibration and estimation.
I will review models proposed in the literature based on multi factor s.d.e. frameworks for commodity modeling, including the basic economic justification for such factors in the models and the different structures proposed. Then based on a recent general class of multi factor s.d.e. models proposed in the literature, we will formulate a special case of such a model for which closed from expressions for futures prices and vanilla options prices may be obtained. This extends the multi-factor long-short model in Schwartz and Smith (Manag Sci 893–911, 2000) and Yan (Review of Derivatives Research 5(3):251–271, 2002) in two important aspects: firstly we allow for both the long and short term dynamic factors to be mean reverting incorporating stochastic volatility factors and secondly we develop an additive structural seasonality model.
A Milstein scheme is used to provide an accurate discretized representation of the s.d.e. model. This results in a challenging non-linear non-Gaussian state-space model. To carry out inference, we develop an adaptive particle Markov chain Monte Carlo method. This methodology allows us to jointly calibrate and filter the latent processes for the long-short and volatility dynamics. This methodology is general and can be applied to the estimation and calibration of many of the other multi-factor stochastic commodity models proposed in the literature. We demonstrate the performance of our model and algorithm on both synthetic data and real data for futures contracts on crude oil.
I will then conclude with some comments on other recent developments in the literature that can extend this class of calibration procedures that can be incorporated under this framework. This includes adaptive strategies in the PMCMC as well as the ability to avoid discretization error whilst allowing for Levy process dynamics in the spot price s.d.e.
The paper is available at arxiv 1105.5850.
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