The main part of the “Commodities’ Products Project“:../ has been developed by researches on the hedging of a commodity-linked portfolio. Lately, it appeared crucial to people coming from the academic or the financial industry to find hedging mechanisms. With the recent development in commodity markets, various authors have analyzed hedging strategies but mostly based on statistical results. However there is no generic analysis of commodity-linked portfolios. Various authors consider portfolios; their studies remain restricted to very particular data. Here we propose a general framework. The purpose is to provide a theoretical support to practical hedging operations. The value-added relies on the accurateness of hedging features starting with a well-established model. We use a benchmark reference: the one-factor future model by Clewlow and Strickland in 1999. In order to validate the approach, we performed practical tests. While working on the study of the hedging, we wrote a paper to be published. The work done so far has been approved by two conferences we will attend in April and May.

The Team

The authors of the paper are:

Assisted for testing and debuging by:

From: ECE Paris Graduate School of Engineering, 37 quai de Grenelle CS71520, 75 725 Paris 15, France

Attended Conferences


Full paper:

Matlab source code:


With the recent developments in commodity markets, it appears to be crucial for people, coming both from the academic and financial industry, to grant some care to the hedging mechanism of a given commodity-linked portfolio. Various authors have considered and analyzed hedging strategies for a commodity long-term position with two shorter futures (see for instance [1, 2, 3, 4]). However it clearly seems there is no generic presentation and analysis for commodity-linked portfolio frameworks. Indeed even if authors consider portfolios their studies remain restricted to very particular data. Their approaches and results are mostly statistic-based. In this project we assume that a commodity-linked portfolio with its related instruments is given. Our purpose is to perform a general analysis for hedging against unfavorable price changes. We will illustrate it by a practical implementation of the initial portfolio. Indeed our main motivation is to provide a theoretical support of a practical hedging tool. The approach relies on a one-factor structure model for the future (or spot prices) as the one introduced by L. Clewlow and C. Strickland [5]. This model can fit the initial forward curve in contrast to the model introduced by Schwartz (1997). It reflects the mean reverting nature of commodity prices. However, it has the disadvantage of the constant volatility structure of forward prices. The use of this restrictive model is justified by the fact that the one-factor uncertainty model remains both in a theoretical and practical point of view as a benchmark reference. Next, it seems to us that this simple framework is suitable to present the main ideas underlying the hedging operation. The case related to another acceptable structure model (as the two-factor one) might also be performed but at the price of technical difficulties. We look forward to do so in a next project. Therefore we first derive the sensitivities of various basic products (futures, swaps, options on spot/future, cap/floor contracts) with respect to the shock responsible for the price changes. The latter is the one that underlies the single factor model under consideration. The point of this work, unlike some classical results, is about the sensitivities nature and the high orders considered. As a matter of fact it is common to make use of sensitivities with respect to the spot/future price or even the uncertainty factor but with limitations to the first order. The drawback of such standard methods is that hedging long maturity positions with short maturity contracts cannot be properly done as illustrated in our various experiments. We will show and illustrate that introducing higher order sensitivities leads to find more accurate hedging operations. Particularly, under a conservative viewpoint of the uncertainty factor, our approach enables to derive deterministic and point-wise estimates of the hedging error. This is economically meaningful in contrast to the standard hedging error in term of variance. The solution we provide for the portfolio-hedging problem has also the advantage of giving the suitable allocation of hedging instruments in term of security numbers, as it is usually required in practice. Contrarily to standard solutions where only proportions of the contracts to use are given, the hedger does not have to make any extra-decision concerning the security quantities. The method considers sensitivities with respect to the shocks that are related to the risk/opportunity factor. That might look a little bit unpleasant to the hedger that is mostly familiar with observable variables. Actually we show that once a high order for sensitivities is chosen the shock levels do not matter since all extreme potential losses/gains are under control and deterministically derived by our approach. In accordance with the real practice, our hedging approach relies on the combination of various instruments that the hedger has at disposal. Only the offsetting effects between the various sensitivities matter here. By contrast, we can observe that several papers dealing with hedging are especially focused on hedging a main given derivative by using associated underlying assets.

Keywords: commodity derivatives, sensitivities, hedging, one factor model

Related works: [6], [7] and [8].

JEL Classification: G11, G13.


  1. C. Alexander, M. Prokopczuck, and A. Sumawong, “The (de)merits of minimum variance hedging: application to the crack spread.” January 2012. [Online]. Available:
  2. O. Korn, “The drift matters: an analysis of commodity futures and options.” May 2002. [Online]. Available:
  3. D. Lautier and A. G. Galli, “Dynamic hedging strategies: an application to the crude oil market.” March 2010. [Online]. Available:
  4. C. D. V. de Goyet, “The performance of the a0(n) diffusion model to hedge a forward commitment in the corn market.” October 2007. [Online]. Available:
  5. L. Clewlow and C. Strickland, “Valuing energy options in a one factor model fitted to forward prices.” April 1999. [Online]. Available:
  6. Y. Rakotondratsimba, “Effect of the asset change on the portfolio return in presence of transaction costs.”
  7. H. Kocherlakota, E. Rosenbloom, and W. Shiu, “Cash-flow matching and linear programming duality” pp. 283-296, 1990.
  8. H. Jaffal, A. Yassine, and Y. Rakotondratsimba, “Enhancement of the bond portfolio immunization under a parallel shift of the yield curve,” 2012.

Go back to the "Commodities' Products Project" page

Go back to the home page