Pierre Six

Dynamic speculation and hedging in commodity futures markets with a stochastic convenience yield

The main objective of this paper is to address, in an a continuous-time framework, the issue of using storable commodity futures as vehicles for hedging purposes when, in particular, the convenience yield as well as the market prices of risk evolve randomly over time. Following the martingale route and by operating a suitable constant relative risk aversion utility function (CRRA) specific change of numeraire, we solve the investors dynamic optimization program to obtain quasi analytical solutions for optimal demands, which can be expressed in terms of two discount bonds (traded and synthetic). Contrary to the existing literature, we explicitly derive the individual optimal proportions invested in the spot commodity, in a discount bond and in the futures contracts, which can be computed in a simple recursive way. We suggest various decompositions allowing an investor to assess the sensitivity of the optimal demands to the state variables and to specify the role played by each risky asset. Empirical evidence shows that the convenience yield has a strong impact on the speculation and hedging positions and the interaction among time-varying risk premia determines the magnitude and the sign of these positions.

Strategic commodity allocation

This article extends the study of the financialization of commodities (Rouwenhorst and Tang [Annu. Rev. Financ. Econ., 2012, 4, 447–467]) by considering an investment in the term structure of commodity futures prices. Specifically, we analyse the benefits of adding a distant commodity futures contract and/or a spot commodity (near month futures contract) to a portfolio of bonds and stocks in a setting similar to Brennan and Schwartz [The use of treasury bill futures in strategic asset allocation programs. In Worldwide Asset and Liability Modeling, edited by W.T. Ziemba and J.M. Mulvey, pp. 205–230, 1998 (Cambridge University Press: Cambridge)]. Our analysis employs an empirical study that covers the post financial crisis period. We show that the spot commodity considerably improves the value of the portfolio. However, an investment in the whole term structure of futures contracts is optimally achieved through high opposite positions in the spot commodity and distant futures contracts. We find that these extreme calendar spreads can result in an inappropriate investment.

On The Shape of Risk Aversion and Asset Allocation

This paper demonstrates the simple incorporation of any shape of risk aversion into an asset allocation framework. Indeed, the relevant literature about risk aversion shows mixed evidence regarding the shape of this important but subjective variable. Our setting builds on, and can be compared with, the well-known constant relative risk aversion (CRRA) framework and mostly preserves the tractability of the affine-CRRA framework. Our numerical analysis exhibits some link between measures of risk aversions and empirical studies of asset allocation.

Correlation as a Pricing Factor for Oil Derivatives

The goal of our paper is to show how correlation between convenience yield and commodity spot price must be and can be integrated to valuate commodity derivatives. This incorporation can be done in addition to usual factors: market prices of risk (Casassus and Collin-Dufresne, 2005) and/or stochastic volatility (Nielsen and Schwartz, 2004; Richter and Sorensen, 2006). While not discussed in this paper for ease of presentation, we could also take into account jumps (Hilliard and Reis, 1998; Koekebakker and Lien, 2004) in addition to stochastic correlation. Indeed, Cuchiero et al. (2011) derive necessary and sufficient conditions for the existence of affine-jump matrix valued processes. Their framework enables the modeling of the above cited stylized facts while keeping the usual tractability of the affine framework to price derivatives as well as to estimate their parameters.

Optimal Dynamic Hedging in Commodity Futures Markets with a Stochastic Convenience Yield

We focus in this article on the impact of the convenience yield on optimal hedging in a futures market. Our investor can freely negotiate the underlying spot commodity and trade in the bond market. We undertake our study in a setting where the three state variables, namely the convenience yield, the spot price and interest rates, as well as the market price of risk evolve randomly over time. We achieve various decompositions of optimal demands to highlight the particular role of each investment instruments regarding the optimal hedge. Despite the thorough description of the risks of the economy, we obtain closed-form solutions, which further facilitate the assessment of the behavior of our investor.

Interest Rate Risk Hedging Demand Under a Gaussian Framework

This article analyzes the state variables Merton-Breeden hedging demand for an investor endowed with a utility function over both intermediate consumption and terminal wealth. Based on the three-factor model of Babbs and Nowman (1999), we show that this demand can be simply expressed as weighted average zero-coupon bonds sensitivities to these factors. The weighting parameter is actually the proportion of wealth our investor sets aside for future consumption rather than for terminal wealth.

Calendar Spreads, Risk Premium and the Convenience Yield

This paper studies optimal calendar spreads in commodity futures markets while taking into account a stochastic convenience yield. We show that a convenience yield imperfectly correlated with the spot commmodity price results in an optimal strategy composed of two commodity futures contracts. These strategies reveal a calendar spread effect through the positive correlation between the two futures contracts. These strategies can easily be computed and analyzed under the Samuelson hypothesis.