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Futures pricing in electricity markets based on stable CARMA spot models


  • Benth, Fred Espen
  • Klüppelberg, Claudia
  • Müller, Gernot
  • Vos, Linda


We present a new model for the electricity spot price dynamics, which is able to capture seasonality, low-frequency dynamics and extreme spikes in the market. Instead of the usual purely deterministic trend we introduce a non-stationary independent increment process for the low-frequency dynamics, and model the large fluctuations by a non-Gaussian stable CARMA process. The model allows for analytic futures prices, and we apply these to model and estimate the whole market consistently. Besides standard parameter estimation, an estimation procedure is suggested, where we fit the non-stationary trend using futures data with long time until delivery, and a robust L1-filter to find the states of the CARMA process. The procedure also involves the empirical and theoretical risk premia which – as a by-product – are also estimated. We apply this procedure to data from the German electricity exchange EEX, where we split the empirical analysis into base load and peak load prices. We find an overall negative risk premium for the base load futures contracts, except for contracts close to delivery, where a small positive risk premium is detected. Peak load contracts, on the other hand, show a clear positive risk premium, when they are close to delivery, while contracts in the longer end also have a negative premium.

Suggested Citation

  • Benth, Fred Espen & Klüppelberg, Claudia & Müller, Gernot & Vos, Linda, 2014. "Futures pricing in electricity markets based on stable CARMA spot models," Energy Economics, Elsevier, vol. 44(C), pages 392-406.
  • Handle: RePEc:eee:eneeco:v:44:y:2014:i:c:p:392-406
    DOI: 10.1016/j.eneco.2014.03.020

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    References listed on IDEAS

    1. Claudia Kluppelberg & Thilo Meyer-Brandis & Andrea Schmidt, 2010. "Electricity spot price modelling with a view towards extreme spike risk," Quantitative Finance, Taylor & Francis Journals, vol. 10(9), pages 963-974.
    2. Eduardo Schwartz & James E. Smith, 2000. "Short-Term Variations and Long-Term Dynamics in Commodity Prices," Management Science, INFORMS, vol. 46(7), pages 893-911, July.
    3. Brockwell, Peter J. & Davis, Richard A. & Yang, Yu, 2011. "Estimation for Non-Negative Lévy-Driven CARMA Processes," Journal of Business & Economic Statistics, American Statistical Association, vol. 29(2), pages 250-259.
    4. Peter J. Brockwell & Richard A. Davis & Yu Yang, 2011. "Estimation for Non-Negative Lévy-Driven CARMA Processes," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 29(2), pages 250-259, April.
    5. Kolos, Sergey P. & Ronn, Ehud I., 2008. "Estimating the commodity market price of risk for energy prices," Energy Economics, Elsevier, vol. 30(2), pages 621-641, March.
    6. P. Brockwell, 2001. "Lévy-Driven Carma Processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(1), pages 113-124, March.
    7. Francis A. Longstaff & Ashley W. Wang, 2004. "Electricity Forward Prices: A High-Frequency Empirical Analysis," Journal of Finance, American Finance Association, vol. 59(4), pages 1877-1900, August.
    8. Hendrik Bessembinder & Michael L. Lemmon, 2002. "Equilibrium Pricing and Optimal Hedging in Electricity Forward Markets," Journal of Finance, American Finance Association, vol. 57(3), pages 1347-1382, June.
    9. Paschke, Raphael & Prokopczuk, Marcel, 2010. "Commodity derivatives valuation with autoregressive and moving average components in the price dynamics," Journal of Banking & Finance, Elsevier, vol. 34(11), pages 2742-2752, November.
    10. Brockwell, Peter J. & Lindner, Alexander, 2009. "Existence and uniqueness of stationary Lévy-driven CARMA processes," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2660-2681, August.
    11. Fred Espen Benth & Jan Kallsen & Thilo Meyer-Brandis, 2007. "A Non-Gaussian Ornstein-Uhlenbeck Process for Electricity Spot Price Modeling and Derivatives Pricing," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(2), pages 153-169.
    12. Davis, Richard A., 1996. "Gauss-Newton and M-estimation for ARMA processes with infinite variance," Stochastic Processes and their Applications, Elsevier, vol. 63(1), pages 75-95, October.
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    Cited by:

    1. Benth, Fred Espen & Koekebakker, Steen, 2015. "Pricing of forwards and other derivatives in cointegrated commodity markets," Energy Economics, Elsevier, vol. 52(PA), pages 104-117.
    2. Fred Espen Benth & Hanna Zdanowicz, 2016. "Pricing And Hedging Of Energy Spread Options And Volatility Modulated Volterra Processes," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(01), pages 1-22, February.
    3. repec:eee:eneeco:v:63:y:2017:i:c:p:51-65 is not listed on IDEAS
    4. Fred Benth & Nils Detering, 2015. "Pricing and hedging Asian-style options on energy," Finance and Stochastics, Springer, vol. 19(4), pages 849-889, October.
    5. repec:eee:eneeco:v:67:y:2017:i:c:p:496-507 is not listed on IDEAS
    6. Awdesch Melzer & Wolfgang K. Härdle & Brenda López Cabrera, 2017. "Pricing Green Financial Products," SFB 649 Discussion Papers SFB649DP2017-020, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    7. Fred Espen Benth & Asma Khedher & Mich`ele Vanmaele, 2017. "Pricing of commodity derivatives on processes with memory," Papers 1711.00307,
    8. Fred Espen Benth & Heidar Eyjolfsson, 2015. "Representation and approximation of ambit fields in Hilbert space," Papers 1509.08272,
    9. Benth, Fred Espen & Paraschiv, Florentina, 2016. "A Structural Model for Electricity Forward Prices," Working Papers on Finance 1611, University of St. Gallen, School of Finance.
    10. repec:eee:eneeco:v:78:y:2019:i:c:p:267-277 is not listed on IDEAS
    11. repec:eee:jbfina:v:95:y:2018:i:c:p:203-216 is not listed on IDEAS
    12. repec:spr:aistmt:v:70:y:2018:i:2:d:10.1007_s10463-017-0601-5 is not listed on IDEAS
    13. Fred Espen Benth & Hanna Zdanowicz, 2014. "Pricing and hedging of energy spread options and volatility modulated Volterra processes," Papers 1409.5801,

    More about this item


    CARMA model; Electricity spot prices; Electricity futures prices; Continuous time linear model; Lévy process; Stable CARMA process; Risk premium; Robust filter;

    JEL classification:

    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing


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