Review of Derivatives Research

, Volume 9, Issue 2, pp 167–186 | Cite as

Seasonal and stochastic effects in commodity forward curves

  • Svetlana Borovkova
  • Helyette Geman
Open Access


In this paper we develop a new model for the dynamics of forward curves of commodities exhibiting seasonalities, such as natural gas, electricity or agricultural commodities. In the existing literature on the subject, the first state variable in multi-factor models is the commodity price, which combines seasonal and stochastic features and may be unobservable. We propose to use instead the average forward price, which is devoid of seasonality and conveys a more robust representation of the current forward curve level. The second factor in the model is a quantity analogous to the stochastic convenience yield, which accounts for the random changes in the forward curve shape. The well-known cost-of-carry relationship is significantly improved by introducing a deterministic seasonal premium within the convenience yield. We develop model estimation procedures and apply them to a number of energy markets.


Commodity futures Forward curve seasonality Energy markets 


  1. Amin, K., Ng, V., & Pirrong, S. C. (1994). Valuing energy derivatives. In Managing energy price risk, Risk Publications.Google Scholar
  2. Black F. (1976). The pricing of commodity contracts. Journal of Financial Economics 3(1/2): 167–179CrossRefGoogle Scholar
  3. Borovkova, S. (2004). The forward curve dynamic and market transition forecasts. In D.W. Bunn (Ed.), Modelling prices in competitive electricity markets. John Wiley & Sons, Ltd. p. 24.Google Scholar
  4. Brennan M.J., Schwartz E.S. (1985). Evaluating natural resource investments. Journal of Business 58(2): 135–157CrossRefGoogle Scholar
  5. Carmona, R., & Ludkovski, M. (2004). Spot convenience yield models for energy markets. In AMS mathematics of finance, G. Yin & Y. Zhang (Eds.), Vol. 351 of Contemporary Mathematics, pp. 65–80.Google Scholar
  6. Eydeland, A., & Geman, H. (1998). Pricing power derivatives. RISK, September 1998, Risk Publications.Google Scholar
  7. Fama E.F., French K.R. (1987). Commodity futures prices: Some evidence on forecast power, premiums and the theory of storage. Journal of Business 60, 55–73CrossRefGoogle Scholar
  8. Geman, H. (2005). Commodities and commodity derivatives. Wiley Finance.Google Scholar
  9. Geman, H. (2007). Mean-reversion versus random walk in energy commodity prices. Advances in Mathematical Finance, Kluwer Publisher.Google Scholar
  10. Geman H., Nguyen V. (2005). Soybean inventory and forward curve dynamics. Management Science 51(7): 1076–1091CrossRefGoogle Scholar
  11. Gibson R., Schwartz E.S. (1990). Stochastic convenience yield and the pricing of oil contingent claims. Journal of Finance 45(3): 959–976CrossRefGoogle Scholar
  12. Heinkel R., Howe M.E., Huges J.S. (1990). Commodity convenience yields as an option profit. Journal of Futures Markets 10(5): 519–533CrossRefGoogle Scholar
  13. Hull J.C., White A. (1990). Pricing interest-rate derivative securities. The Review of Financial Studies 3(4): 573–592CrossRefGoogle Scholar
  14. Kaldor N. (1939). Speculation and economic stability. Review of Economic Studies 7, 1–27CrossRefGoogle Scholar
  15. Litzenberger R.H., Rabinowitz N. (1995). Backwardation in oil futures markets: Theory and empirical evidence. Journal of Finance 50(5): 1517–1545CrossRefGoogle Scholar
  16. Lucia J., Schwartz E.S. (2002). Electricity prices and power derivatives: Evidence from the nordic power exchange, Review of Derivatives Research 5(1): 5–50Google Scholar
  17. Merton R.C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science 4(1): 141–183CrossRefGoogle Scholar
  18. Milonas N.T. (1991). Measuring seasonalities. In Commodity markets and the half-month effect. Journal of Futures Markets 11(3): 331–346Google Scholar
  19. Routledge B.R., Seppi D.J., Spatt C.S. (2000). Equilibrium forward curves for commodities. Journal of Finance 55(3): 1297–1338CrossRefGoogle Scholar
  20. Schwartz E.S. (1997). The stochastic behaviour of commodity prices: Implications for valuation and hedging. Journal of Finance 53(3): 923–973CrossRefGoogle Scholar
  21. Sorensen C. (2002). Modeling seasonality in agricultural commodity futures. Journal of Futures Markets 22(5): 393–426CrossRefGoogle Scholar
  22. Wooldridge, J. M. (1999). Introductory econometrics: A modern approach. South-Western College Publishing.Google Scholar
  23. Working H. (1948). The theory of price of storage. Journal of Farm Economics 30, 1–28CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Finance, Faculty of EconomicsFree University of AmsterdamAmsterdamThe Netherlands
  2. 2.Birkbeck, University of LondonLondonUK
  3. 3.ESSEC Business SchoolCergy-Pontoise CedexFrance

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