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On the speed towards the mean for continuous time autoregressive moving average processes with applications to energy markets

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  • Benth, Fred Espen
  • Taib, Che Mohd Imran Che

Abstract

We extend the concept of half life of an Ornstein–Uhlenbeck process to Lévy-driven continuous-time autoregressive moving average processes with stochastic volatility. The half life becomes state dependent, and we analyze its properties in terms of the characteristics of the process. An empirical example based on daily temperatures observed in Petaling Jaya, Malaysia, is presented, where the proposed model is estimated and the distribution of the half life is simulated. The stationarity of the dynamics yield futures prices which asymptotically tend to constant at an exponential rate when time to maturity goes to infinity. The rate is characterized by the eigenvalues of the dynamics. An alternative description of this convergence can be given in terms of our concept of half life.

Suggested Citation

  • Benth, Fred Espen & Taib, Che Mohd Imran Che, 2013. "On the speed towards the mean for continuous time autoregressive moving average processes with applications to energy markets," Energy Economics, Elsevier, vol. 40(C), pages 259-268.
    • Handle: RePEc:eee:eneeco:v:40:y:2013:i:c:p:259-268
    DOI: 10.1016/j.eneco.2013.07.007
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    References listed on IDEAS

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    1. Fred Espen Benth & Jurate Saltyte-Benth, 2005. "Stochastic Modelling of Temperature Variations with a View Towards Weather Derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 12(1), pages 53-85.
    2. Fred Espen Benth & Jūratė Šaltytė Benth & Steen Koekebakker, 2007. "Putting a Price on Temperature," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 34(4), pages 746-767, December.
    3. 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.
    4. Dorfleitner, Gregor & Wimmer, Maximilian, 2010. "The pricing of temperature futures at the Chicago Mercantile Exchange," Journal of Banking & Finance, Elsevier, vol. 34(6), pages 1360-1370, June.
    5. Sean D. Campbell & Francis X. Diebold, 2005. "Weather Forecasting for Weather Derivatives," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 6-16, March.
    6. Härdle, Wolfgang Karl & López Cabrera, Brenda, 2009. "Implied market price of weather risk," SFB 649 Discussion Papers 2009-001, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    7. 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.
    8. 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.
    9. Peter Alaton & Boualem Djehiche & David Stillberger, 2002. "On modelling and pricing weather derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 9(1), pages 1-20.
    10. Ole E. Barndorff-Nielsen & Fred Espen Benth & Almut E. D. Veraart, 2013. "Modelling energy spot prices by volatility modulated L\'{e}vy-driven Volterra processes," Papers 1307.6332, arXiv.org.
    11. 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.
    12. Wolfgang Karl Härdle & Brenda López Cabrera, 2012. "The Implied Market Price of Weather Risk," Applied Mathematical Finance, Taylor & Francis Journals, vol. 19(1), pages 59-95, February.
    13. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
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    More about this item

    Keywords

    CARMA processes; Stationarity; Half life; Mean reversion;
    All these keywords.

    JEL classification:

    • Q40 - Agricultural and Natural Resource Economics; Environmental and Ecological Economics - - Energy - - - General
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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