Arbitrage free cointegrated models in gas and oil future markets

With the energy market liberalisation over the last decade, one may think that gas and oil prices are decoupled, but actually several statistical studies tend to prove that prices are cointegrated. For statistical evidences of cointegration and economic explanations, we refer for instance to the articles by Panagiotidis and Rutledge [15] and Asche et al. [1] for the UK and European markets, and to the article by Bachmeier and Griffin [2] for the US market. The dependence between gas and oil prices could be economically explained with gas long term contracts, which still represent the majority of supply in European gas and whose prices are indexed on oil and oil products prices. This indexation creates a structural link between prices of both energies.

In the previously cited references, one proposes econometric models (in discrete time) for gas/oil prices, which are coherent with market data and which accounts well for the interdependence of prices. They are useful for some risk management purposes, such as Value at Risk measurements. However, as soon as we have to consider energy contracts and related pricing/hedging issues, different models emerge in order to be consistent with the arbitrage free theory: they are such that forward contracts are martingales under risk neutral probabilities (see Musiela and Rutkowski [13]). In the following and to simplify our presentation, we identify future prices (given by quotations data) and forward prices (given by models), which is correct if interest rates are deterministic for instance. Usual factor models for the spot and forward contracts on a given energy are written as dF (t, T )

where

• F (t, T ) is the forward contract quoted in t and delivered in T ,

• σ(t, T ) = (σ 1 (t, T ), σ 2 (t, T ), . . . , σ n (t, T )) is a row vector of normalized volatility functions of forward returns, • n is the number of risk factors identified through a PCA (Principal Component Analysis) of forward returns,

• are independant Brownians motions under risk neutral probabilities (here * stands for the transposition),

• Σ is a n×n matrix, equal to the square root of a variance-covariance matrix.

See the works by Geman [9], Clewlow and Strickland [6] among others.

The volatility functions describe the shifting, the twisting and the bending (for Clewlow and Strickland [6]). In Brooks [4], they describe the level, the slope, the curvature (LSC model). R. Brooks uses this model for natural gas contracts and gives an explicit form for these functions :

The model is linear in the parameters (Σ and (τ e i ) i ), thus ordinary least squares regression is applied to estimate them.

For crude oil contracts, analogous models could be set up, with different Brownian motions. These ones can be correlated to those of gas models. Within this approach, it appears that volatility functions adjust well for each energy. Nevertheless, long term dependences are poorly modeled. The variance-covariance matrix (related to Σ) induces relevant marginal distributions of the returns but unfortunately unrealistic joint price distribution. For example, when simulating the model one often obtains a growing prices’ scenario for gas and a decreasing one for crude oil (see Figure 4 in Section 3). It is fundamental to note that the previously decribed models (1) are written under the risk-neutral probability (denoted by Q in the sequel), which is suitable for pricing/hedging issues, while the long-term dependence (given by an econometric cointegration analysis) holds under the historical (or physical) probability (denoted by P). To accommodate both features (long term dependences on the one hand; stochastic returns described by volatility functions on the other hand), a natural idea consists in suitably modeling the market price of risk (λ t ) t , making the connection between historical and risk-neutral worlds. This is the main contribution of our work. Details are given in Section 2. Thus, our model cointegrates gas and oil prices while being coherent with the pricing by arbitrage. In the following, we mainly focus on natural gas and crude oil.

We now mention a similar approach to ours. In his PhD thesis, Steve Ohana aims at modeling spot and forward contracts for two cointegrated energies, but in discrete time (see Ohana [14]). He handles the case of US natural gas and crude oil market. For each energy e (e = g for gas, e = c for crude), the model is given by (under P) Finally we mention very recent works, whose main motivations are to analyse the market risk premium π e (t, T ) through the specification of the market price of risk (λ t ) t . We recall that th

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