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A mean reverting affine GARCH model for commodities

Author

Listed:
  • Escobar-Anel, Marcos
  • Pan, Kaize
  • Stentoft, Lars

Abstract

We introduce a new discrete time model for commodity spot prices that integrates mean reversion with time varying volatility. Because of its affine structure, the model delivers closed-form moment-generating functions for spot prices and, with appropriate risk neutralization, a mean-reverting affine representation for future prices under both the historical and risk-neutral measures. This permits analytical pricing of derivatives on the spot and futures contracts. Estimation can be done with maximum likelihood, the reliability of which is confirmed in a simulation study using spot prices and spot and futures prices jointly. An empirical application reveals significant mean reversion and time-varying conditional variance in three selected commodities, and shows that using models without mean reversion or time varying volatility results in much lower fit in terms of likelihood and much larger futures pricing errors both in- and out-of-sample.

Suggested Citation

  • Escobar-Anel, Marcos & Pan, Kaize & Stentoft, Lars, 2026. "A mean reverting affine GARCH model for commodities," Energy Economics, Elsevier, vol. 153(C).
    • Handle: RePEc:eee:eneeco:v:153:y:2026:i:c:s0140988325009053
    DOI: 10.1016/j.eneco.2025.109075
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    References listed on IDEAS

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    1. Crosby, John & Frau, Carme, 2022. "Jumps in commodity prices: New approaches for pricing plain vanilla options," Energy Economics, Elsevier, vol. 114(C).
    2. Peter Christoffersen & Steven Heston & Kris Jacobs, 2013. "Capturing Option Anomalies with a Variance-Dependent Pricing Kernel," The Review of Financial Studies, Society for Financial Studies, vol. 26(8), pages 1963-2006.
    3. Christoffersen, Peter & Jacobs, Kris & Ornthanalai, Chayawat, 2012. "Dynamic jump intensities and risk premiums: Evidence from S&P500 returns and options," Journal of Financial Economics, Elsevier, vol. 106(3), pages 447-472.
    4. Cortazar, Gonzalo & Schwartz, Eduardo S., 2003. "Implementing a stochastic model for oil futures prices," Energy Economics, Elsevier, vol. 25(3), pages 215-238, May.
    5. Escobar-Anel, Marcos & Rastegari, Javad & Stentoft, Lars, 2021. "Option pricing with conditional GARCH models," European Journal of Operational Research, Elsevier, vol. 289(1), pages 350-363.
    6. Narayan, Paresh Kumar & Liu, Ruipeng, 2015. "A unit root model for trending time-series energy variables," Energy Economics, Elsevier, vol. 50(C), pages 391-402.
    7. Zhang, Yongmin & Wang, Ruizhi, 2022. "COVID-19 impact on commodity futures volatilities," Finance Research Letters, Elsevier, vol. 47(PA).
    8. Xinyu Yuan & Jiechen Tang & Wing-Keung Wong & Songsak Sriboonchitta, 2020. "Modeling Co-Movement among Different Agricultural Commodity Markets: A Copula-GARCH Approach," Sustainability, MDPI, vol. 12(1), pages 1-17, January.
    9. Schwartz, Eduardo S, 1997. "The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging," Journal of Finance, American Finance Association, vol. 52(3), pages 923-973, July.
    10. Escobar-Anel, Marcos & Stentoft, Lars & Ye, Xize, 2025. "Analytical fixed income pricing in discrete time: A new family of models," Global Finance Journal, Elsevier, vol. 67(C).
    11. Li, Bingxin, 2019. "Pricing dynamics of natural gas futures," Energy Economics, Elsevier, vol. 78(C), pages 91-108.
    12. Heston, Steven L & Nandi, Saikat, 2000. "A Closed-Form GARCH Option Valuation Model," The Review of Financial Studies, Society for Financial Studies, vol. 13(3), pages 585-625.
    13. Tianyi Wang & Sicong Cheng & Fangsheng Yin & Mei Yu, 2022. "Directly pricing VIX futures: the role of dynamic volatility and jump intensity," Applied Economics, Taylor & Francis Journals, vol. 54(32), pages 3678-3694, July.
    14. Katherine B. Ensor & Yu Han & Barbara Ostdiek & Stuart M. Turnbull, 2020. "Dynamic jump intensities and news arrival in oil futures markets," Journal of Asset Management, Palgrave Macmillan, vol. 21(4), pages 292-325, July.
    15. Wong, Hoi Ying & Lo, Yu Wai, 2009. "Option pricing with mean reversion and stochastic volatility," European Journal of Operational Research, Elsevier, vol. 197(1), pages 179-187, August.
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    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection

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