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On Finite Dimensional Realizations For The Term Structure Of Futures Prices

Author

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  • TOMAS BJÖRK

    (Department of Finance, Stockholm School of Economics, Box 6501, SE-113 83 Stockholm, Sweden)

  • MAGNUS BLIX

    (Department of Finance, Stockholm School of Economics, Box 6501, SE-113 83 Stockholm, Sweden)

  • CAMILLA LANDÉN

    (Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden)

Abstract

We consider HJM type models for the term structure of futures prices, where the volatility is allowed to be an arbitrary smooth functional of the present futures price curve. Using a Lie algebraic approach we investigate when the infinite dimensional futures price process can be realized by a finite dimensional Markovian state space model, and we give general necessary and sufficient conditions, in terms of the volatility structure, for the existence of a finite dimensional realization. We study a number of concrete applications including a recently developed model for gas futures. In particular we provide necessary and sufficient conditions for when the induced spot price is a Markov process. In particular we can prove that the only HJM type futures price models with spot price dependent volatility structures which generically possess a spot price realization are the affine ones. These models are thus the only generic spot price models from a futures price term structure point of view.

Suggested Citation

  • Tomas Björk & Magnus Blix & Camilla Landén, 2006. "On Finite Dimensional Realizations For The Term Structure Of Futures Prices," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 9(03), pages 281-314.
    • Handle: RePEc:wsi:ijtafx:v:09:y:2006:i:03:n:s0219024906003639
    DOI: 10.1142/S0219024906003639
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    1. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    2. Tomas Björk & Lars Svensson, 2001. "On the Existence of Finite‐Dimensional Realizations for Nonlinear Forward Rate Models," Mathematical Finance, Wiley Blackwell, vol. 11(2), pages 205-243, April.
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    7. Camilla Landén & Tomas Björk, 2002. "On the construction of finite dimensional realizations for nonlinear forward rate models," Finance and Stochastics, Springer, vol. 6(3), pages 303-331.
    8. Björk, Tomas & Landen, Camilla, 2000. "On the Term Structure of Futures and Forward Prices," SSE/EFI Working Paper Series in Economics and Finance 0417, Stockholm School of Economics, revised 20 Dec 2000.
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    Cited by:

    1. Chiarella, Carl & Kang, Boda & Nikitopoulos, Christina Sklibosios & Tô, Thuy-Duong, 2013. "Humps in the volatility structure of the crude oil futures market: New evidence," Energy Economics, Elsevier, vol. 40(C), pages 989-1000.
    2. Leif Andersen, 2010. "Markov models for commodity futures: theory and practice," Quantitative Finance, Taylor & Francis Journals, vol. 10(8), pages 831-854.
    3. Fred Benth & Jukka Lempa, 2014. "Optimal portfolios in commodity futures markets," Finance and Stochastics, Springer, vol. 18(2), pages 407-430, April.
    4. Cheng, Benjamin & Nikitopoulos, Christina Sklibosios & Schlögl, Erik, 2018. "Pricing of long-dated commodity derivatives: Do stochastic interest rates matter?," Journal of Banking & Finance, Elsevier, vol. 95(C), pages 148-166.
    5. Raquel M. Gaspar & Mariana Khapko, 2023. "In memoriam: Tomas Björk (1947–2021)," Finance and Stochastics, Springer, vol. 27(4), pages 867-885, October.

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    More about this item

    Keywords

    HJM models; factor models; state space models; Markovian realizations;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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