IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2609.24548.html

Affine Volterra covariance processes and application to commodity markets

Author

Listed:
  • Boris Gunther
  • Ludger Overbeck

Abstract

We study affine stochastic Volterra equations on the cone of symmetric positive semidefinite matrices. For scalar kernels acting entrywise on the matrix dynamics, we establish weak existence by exploiting stochastic invariance results for Volterra equations on convex domains and derive a conditional Fourier--Laplace transform formula characterized by matrix-valued Riccati--Volterra equations. As an application, we extend the Gibson--Schwartz commodity model by replacing its variance-covariance structure with a Volterra--Wishart process. The resulting model allows for memory in the variances and for stochastic instantaneous correlation, while retaining affine tractability. Its joint Fourier--Laplace transform admits an exponential-affine representation governed by a matrix Riccati--Volterra equation. While the existence theory considered here excludes kernels that are singular at the origin, shifted fractional kernels remain admissible and provide a tractable specification with power-law memory.

Suggested Citation

  • Boris Gunther & Ludger Overbeck, 2026. "Affine Volterra covariance processes and application to commodity markets," Papers 2609.24548, arXiv.org.
    • Handle: RePEc:arx:papers:2609.24548
    as

    Download full text from publisher

    File URL: https://arxiv.org/pdf/2609.24548
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Francis X. Diebold & Kamil Yilmaz, 2009. "Measuring Financial Asset Return and Volatility Spillovers, with Application to Global Equity Markets," Economic Journal, Royal Economic Society, vol. 119(534), pages 158-171, January.
    2. Diebold, Francis X. & Yilmaz, Kamil, 2012. "Better to give than to receive: Predictive directional measurement of volatility spillovers," International Journal of Forecasting, Elsevier, vol. 28(1), pages 57-66.
    3. Samuel Hikspoors & Sebastian Jaimungal, 2008. "Asymptotic Pricing of Commodity Derivatives using Stochastic Volatility Spot Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 15(5-6), pages 449-477.
    4. Chen, Jilong & Ewald, Christian-Oliver, 2017. "Pricing commodity futures options in the Schwartz multi factor model with stochastic volatility: An asymptotic method," International Review of Financial Analysis, Elsevier, vol. 52(C), pages 144-151.
    5. Stefan Trück & Rafał Weron, 2016. "Convenience Yields and Risk Premiums in the EU‐ETS—Evidence from the Kyoto Commitment Period," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 36(6), pages 587-611, June.
    6. Bennedsen, Mikkel, 2017. "A rough multi-factor model of electricity spot prices," Energy Economics, Elsevier, vol. 63(C), pages 301-313.
    7. Christa Cuchiero & Damir Filipovi'c & Eberhard Mayerhofer & Josef Teichmann, 2009. "Affine processes on positive semidefinite matrices," Papers 0910.0137, arXiv.org, revised Apr 2011.
    8. W. Keener Hughen, 2010. "A maximal affine stochastic volatility model of oil prices," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 30(2), pages 101-133, February.
    9. Richter, Martin & Sørensen, Carsten, 2002. "Stochastic Volatility and Seasonality in Commodity Futures and Options: The Case of Soybeans," Working Papers 2002-4, Copenhagen Business School, Department of Finance.
    10. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    11. Christian Gourieroux & Razvan Sufana, 2003. "Whishart Quadratic Term Structure Models," Working Papers 2003-50, Center for Research in Economics and Statistics.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Max F. Schöne & Stefan Spinler, 2017. "A four-factor stochastic volatility model of commodity prices," Review of Derivatives Research, Springer, vol. 20(2), pages 135-165, July.
    2. repec:uts:finphd:41 is not listed on IDEAS
    3. Richter, Anja, 2014. "Explicit solutions to quadratic BSDEs and applications to utility maximization in multivariate affine stochastic volatility models," Stochastic Processes and their Applications, Elsevier, vol. 124(11), pages 3578-3611.
    4. Mesias Alfeus, 2019. "Stochastic Modelling of New Phenomena in Financial Markets," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2019, January-A.
    5. Gonzalo Cortazar & Simon Gutierrez & Hector Ortega, 2016. "Empirical Performance of Commodity Pricing Models: When is it Worthwhile to Use a Stochastic Volatility Specification?," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 36(5), pages 457-487, May.
    6. Seiler, Volker, 2024. "The relationship between Chinese and FOB prices of rare earth elements – Evidence in the time and frequency domain," The Quarterly Review of Economics and Finance, Elsevier, vol. 95(C), pages 160-179.
    7. Rainer Baule & Bart Frijns & Milena E. Tieves, 2018. "Volatility discovery and volatility quoting on markets for options and warrants," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 38(7), pages 758-774, July.
    8. Jaeck, Edouard & Lautier, Delphine, 2016. "Volatility in electricity derivative markets: The Samuelson effect revisited," Energy Economics, Elsevier, vol. 59(C), pages 300-313.
    9. Cortazar, Gonzalo & Lopez, Matias & Naranjo, Lorenzo, 2017. "A multifactor stochastic volatility model of commodity prices," Energy Economics, Elsevier, vol. 67(C), pages 182-201.
    10. Shi, Huai-Long & Zhou, Wei-Xing, 2022. "Factor volatility spillover and its implications on factor premia," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 80(C).
    11. Su, Tong & Lin, Boqiang, 2024. "Reassessing the information transmission and pricing influence of Shanghai crude oil futures: A time-varying perspective," Energy Economics, Elsevier, vol. 140(C).
    12. Mushtaq, Hina & Ishtiaq, Muhammad & Jamal, Surayya & Raza Rizvi, Syed Maisam & Raza, Hamad, 2026. "Regime-Switching volatility and risk quantification in South Asian and developed stock Markets: A Comparative perspective using Markov-Switching GARCH with MLE and MCMC estimations," The North American Journal of Economics and Finance, Elsevier, vol. 82(C).
    13. Tihana Škrinjarić, 2019. "Time Varying Spillovers between the Online Search Volume and Stock Returns: Case of CESEE Markets," IJFS, MDPI, vol. 7(4), pages 1-30, October.
    14. Claeys, Peter & Vašíček, Bořek, 2014. "Measuring bilateral spillover and testing contagion on sovereign bond markets in Europe," Journal of Banking & Finance, Elsevier, vol. 46(C), pages 151-165.
    15. Juncal Cunado & David Gabauer & Rangan Gupta, 2024. "Realized volatility spillovers between energy and metal markets: a time-varying connectedness approach," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 10(1), pages 1-17, December.
    16. Aysan, Ahmet Faruk & Batten, Jonathan & Gozgor, Giray & Khalfaoui, Rabeh & Nanaeva, Zhamal, 2024. "Metaverse and financial markets: A quantile-time-frequency connectedness analysis," Research in International Business and Finance, Elsevier, vol. 72(PB).
    17. Christian Urom & Gideon Ndubuisi & Jude Ozor, 2021. "Economic activity, and financial and commodity markets’ shocks: An analysis of implied volatility indexes," International Economics, CEPII research center, issue 165, pages 51-66.
    18. Duc Huynh, Toan Luu & Burggraf, Tobias & Nasir, Muhammad Ali, 2020. "Financialisation of natural resources & instability caused by risk transfer in commodity markets," Resources Policy, Elsevier, vol. 66(C).
    19. Edib Smolo & Rashed Jahangir & Ruslan Nagayev & Christo S. C. Tarazi, 2022. "Performances of Islamic and conventional equities during the global health crisis: Time‐frequency analysis of BRICS+T markets," Review of Financial Economics, John Wiley & Sons, vol. 40(3), pages 259-280, July.
    20. Shiyuan Li & Xin Li, 2025. "Spillover effects of climate transition risk and financial sectors: New evidence from China," Australian Economic Papers, Wiley Blackwell, vol. 64(1), pages 71-90, March.
    21. Akhtaruzzaman, Md & Boubaker, Sabri & Sensoy, Ahmet, 2021. "Financial contagion during COVID–19 crisis," Finance Research Letters, Elsevier, vol. 38(C).

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2609.24548. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: https://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.