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Minimal realizations of interest rate models

Author

Listed:
  • Tomas BjÃrk

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

  • Andrea Gombani

    (LADSEB-CNR, Corso Stati Uniti 4, I-35127 Padova, Italy Manuscript)

Abstract

We consider interest rate models where the forward rates are allowed to be driven by a multidimensional Wiener process as well as by a marked point process. Assuming a deterministic volatility structure, and using ideas from systems and control theory, we investigate when the input-output map generated by such a model can be realized by a finite dimensional stochastic differential equation. We give necessary and sufficient conditions, in terms of the given volatility structure, for the existence of a finite dimensional realization and we provide a formula for the determination of the dimension of a minimal realization. The abstract state space for a minimal realization is shown to have an immediate economic interpretation in terms of a minimal set of benchmark forward rates, and we give explicit formulas for bond prices in terms of the benchmark rates as well as for the computation of derivative prices.

Suggested Citation

  • Tomas BjÃrk & Andrea Gombani, 1999. "Minimal realizations of interest rate models," Finance and Stochastics, Springer, vol. 3(4), pages 413-432.
    • Handle: RePEc:spr:finsto:v:3:y:1999:i:4:p:413-432
    Note: received: July 1997; final version received: December 1998
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    Citations

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    Cited by:

    1. Haitao Li & Xiaoxia Ye, 2013. "A Type of HJM Based Affine Model: Theory and Empirical Evidence," Working Papers 2013-10-14, Wang Yanan Institute for Studies in Economics (WISE), Xiamen University.
    2. Björk, Tomas, 2003. "On the Geometry of Interest Rate Models," SSE/EFI Working Paper Series in Economics and Finance 545, Stockholm School of Economics.
    3. Eckhard Platen & Stefan Tappe, 2011. "Affine Realizations for Levy Driven Interest Rate Models with Real-World Forward Rate Dynamics," Research Paper Series 289, Quantitative Finance Research Centre, University of Technology, Sydney.
    4. Björk, Tomas, 2000. "A Geometric View of Interest Rate Theory," SSE/EFI Working Paper Series in Economics and Finance 419, Stockholm School of Economics, revised 21 Dec 2000.
    5. 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.
    6. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742.
    7. Björk, Tomas & Landén, Camilla & Svensson, Lars, 2002. "Finite dimensional Markovian realizations for stochastic volatility forward rate models," SSE/EFI Working Paper Series in Economics and Finance 498, Stockholm School of Economics, revised 07 May 2002.
    8. Fred Benth & Jukka Lempa, 2014. "Optimal portfolios in commodity futures markets," Finance and Stochastics, Springer, vol. 18(2), pages 407-430, April.
    9. Carl Chiarella & Oh Kang Kwon, 2001. "Forward rate dependent Markovian transformations of the Heath-Jarrow-Morton term structure model," Finance and Stochastics, Springer, vol. 5(2), pages 237-257.
    10. Andrea Gombani & Wolfgang J. Runggaldier, 2001. "A Filtering Approach To Pricing In Multifactor Term Structure Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 4(02), pages 303-320.
    11. Raquel M. Gaspar & Mariana Khapko, 2023. "In memoriam: Tomas Björk (1947–2021)," Finance and Stochastics, Springer, vol. 27(4), pages 867-885, October.
    12. Stefan Tappe, 2019. "An alternative approach on the existence of affine realizations for HJM term structure models," Papers 1907.03256, arXiv.org.
    13. Stefan Tappe, 2019. "Existence of affine realizations for L\'evy term structure models," Papers 1907.02363, arXiv.org.
    14. Leitner, Johannes, 2000. "Convergence of Arbitrage-free Discrete Time Markovian Market Models," CoFE Discussion Papers 00/07, University of Konstanz, Center of Finance and Econometrics (CoFE).
    15. Claudio Fontana & Giacomo Lanaro & Agatha Murgoci, 2024. "The geometry of multi-curve interest rate models," Papers 2401.11619, arXiv.org, revised Jun 2024.
    16. Gombani, Andrea & Jaschke, Stefan R. & Runggaldier, Wolfgang J., 2005. "A filtered no arbitrage model for term structures from noisy data," Stochastic Processes and their Applications, Elsevier, vol. 115(3), pages 381-400, March.
    17. Fred Espen Benth & Paul Kruhner, 2014. "Representation of infinite dimensional forward price models in commodity markets," Papers 1403.4111, arXiv.org.
    18. Carl Chiarella & Oh-Kang Kwon, 2001. "State Variables and the Affine Nature of Markovian HJM Term Structure Models," Research Paper Series 52, Quantitative Finance Research Centre, University of Technology, Sydney.
    19. Li, Haitao & Ye, Xiaoxia & Yu, Fan, 2020. "Unifying Gaussian dynamic term structure models from a Heath–Jarrow–Morton perspective," European Journal of Operational Research, Elsevier, vol. 286(3), pages 1153-1167.
    20. Likuan Qin & Vadim Linetsky, 2018. "Long-term factorization in Heath–Jarrow–Morton models," Finance and Stochastics, Springer, vol. 22(3), pages 621-641, July.
    21. C. D. D. Neumann, 2007. "On the structure of Gaussian pricing models and Gaussian Markov functional models," Quantitative Finance, Taylor & Francis Journals, vol. 7(5), pages 487-496.
    22. Duffie, Darrell, 2003. "Intertemporal asset pricing theory," Handbook of the Economics of Finance, in: G.M. Constantinides & M. Harris & R. M. Stulz (ed.), Handbook of the Economics of Finance, edition 1, volume 1, chapter 11, pages 639-742, Elsevier.

    More about this item

    Keywords

    Interest rates; realization theory; factor models;
    All these keywords.

    JEL classification:

    • E43 - Macroeconomics and Monetary Economics - - Money and Interest Rates - - - Interest Rates: Determination, Term Structure, and Effects
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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