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Minimal realizations of interest rate models

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  • 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.

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Bibliographic Info

Article provided by Springer in its journal Finance and Stochastics.

Volume (Year): 3 (1999)
Issue (Month): 4 ()
Pages: 413-432

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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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Related research

Keywords: Interest rates; realization theory; factor models;

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Cited by:
  1. Björk, Tomas, 2000. "A Geometric View of Interest Rate Theory," Working Paper Series in Economics and Finance 419, Stockholm School of Economics, revised 21 Dec 2000.
  2. Fred Espen Benth & Jukka Lempa, 2012. "Optimal portfolios in commodity futures markets," Papers 1204.2667, arXiv.org.
  3. Fred Espen Benth & Paul Kr\"uhner, 2014. "Representation of infinite dimensional forward price models in commodity markets," Papers 1403.4111, arXiv.org.
  4. 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.
  5. Fred Benth & Jukka Lempa, 2014. "Optimal portfolios in commodity futures markets," Finance and Stochastics, Springer, vol. 18(2), pages 407-430, April.
  6. repec:wyi:wpaper:002014 is not listed on IDEAS
  7. Johannes Leitner, 2000. "Convergence of Arbitrage-free Discrete Time Markovian Market Models," CoFE Discussion Paper 00-07, Center of Finance and Econometrics, University of Konstanz.
  8. 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.
  9. Björk, Tomas & Landén, Camilla & Svensson, Lars, 2002. "Finite dimensional Markovian realizations for stochastic volatility forward rate models," Working Paper Series in Economics and Finance 498, Stockholm School of Economics, revised 06 May 2002.
  10. Björk, Tomas, 2003. "On the Geometry of Interest Rate Models," Working Paper Series in Economics and Finance 545, Stockholm School of Economics.
  11. 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.
  12. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742, September.
  13. Björk, Tomas & Landen, Camilla, 2000. "On the construction of finite dimensional realizations for nonlinear forward rate models," Working Paper Series in Economics and Finance 420, Stockholm School of Economics.

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