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Abstract-In this paper, a pricing model is proposed for cointegrated commodities extending Larsson model. The futures formulae have been derived considering a linear combination of a Brownian motion and an Ornstein-Uhlenbeck process describing the co-integration relationship of the different futures prices commodities. Tests have been performed with a non-constant volatility in order to fit better the real behavior of the volatility. The model has been applied to energy commodities (gas, CO2, energy) and soft commodities (corn, wheat). Results show that first, the model can be used with different kind of commodities at the cost of a proper parameters calibration and in second, using a non-constant volatility leads to more accurate short term prices, which provides better evaluation of Value-at-Risk and more generally improves the risk management.

This article develops a model that takes into account skewness risk in risk parity portfolios. In this framework, asset returns are viewed as stochastic processes with jumps or random variables generated by a Gaussian mixture distribution. This dual representation allows us to show that skewness and jump risks are equivalent. As the mixture representation is simple, we obtain analytical formulas for computing asset risk contributions of a given portfolio. Therefore, we define risk budgeting portfolios and derive existence and uniqueness conditions. We then apply our model to the equity/bond/volatility asset mix policy. When assets exhibit jump risks like the short volatility strategy, we show that skewness-based risk parity portfolios produce better allocation than volatility-based risk parity portfolios. Finally, we illustrate how this model is suitable to manage the skewness risk of long-only equity factor portfolios and to allocate between alternative risk premia.