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On the Existence of Finite Dimensional Realizations for Nonlinear Forward Rate Models

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Author Info

  • Björk, Tomas

    ()
    (Dept. of Finance, Stockholm School of Economics)

  • Svensson, Lars

    ()
    (Department of Mathematics)

Abstract

We consider interest rate models of Heath-Jarrow-Morton type, where the forward rates are driven by a multidimensional Wiener process, and where the volatility is allowed to be an arbitrary smooth functional of the present forward rate curve. Using ideas from differential geometry as well as from systems and control theory, we investigate when the forward rate 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. A number of concrete applications are given, and most previously known realization results for time homogenous Wiener driven models are included and extended. As a special case we give a general and easily applicable necessary and sufficient condition for when the induced short rate is a Markov porcess. In particular we show that the only forward rate models, with short rate dependent volatility structures, which generically give rise to a Markovian short rate are the affine ones. These models are thus the only generic short rate models from a forward rate point of view.

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Bibliographic Info

Paper provided by Stockholm School of Economics in its series Working Paper Series in Economics and Finance with number 338.

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Length: 46 pages
Date of creation: 22 Oct 1999
Date of revision:
Publication status: Published in Mathematical Finance, 2001, pages 205-243.
Handle: RePEc:hhs:hastef:0338

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Related research

Keywords: Forward rate curves; interest rate models; factor models; state space models; Markovian realizations;

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References

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  1. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, Elsevier, vol. 5(2), pages 177-188, November.
  2. Björk, Tomas & Landen, Camilla, 2000. "On the construction of finite dimensional realizations for nonlinear forward rate models," Working Paper Series in Economics and Finance, Stockholm School of Economics 420, Stockholm School of Economics.
  3. Björk, Tomas & Gombani, Andrea, 1997. "Minimal Realizations of Forward Rates," Working Paper Series in Economics and Finance, Stockholm School of Economics 182, Stockholm School of Economics.
  4. Nelson, Charles R & Siegel, Andrew F, 1987. "Parsimonious Modeling of Yield Curves," The Journal of Business, University of Chicago Press, University of Chicago Press, vol. 60(4), pages 473-89, October.
  5. Ram Bhar & Carl Chiarella, 1995. "Transformation of Heath-Jarrow-Morton Models to Markovian Systems," Working Paper Series, Finance Discipline Group, UTS Business School, University of Technology, Sydney 53, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
  6. Hull, John & White, Alan, 1990. "Pricing Interest-Rate-Derivative Securities," Review of Financial Studies, Society for Financial Studies, Society for Financial Studies, vol. 3(4), pages 573-92.
  7. Carl Chiarella & Oh Kang Kwon, 2001. "Forward rate dependent Markovian transformations of the Heath-Jarrow-Morton term structure model," Finance and Stochastics, Springer, Springer, vol. 5(2), pages 237-257.
  8. Inui, Koji & Kijima, Masaaki, 1998. "A Markovian Framework in Multi-Factor Heath-Jarrow-Morton Models," Journal of Financial and Quantitative Analysis, Cambridge University Press, Cambridge University Press, vol. 33(03), pages 423-440, September.
  9. Vasicek, Oldrich Alfonso, 1977. "Abstract: An Equilibrium Characterization of the Term Structure," Journal of Financial and Quantitative Analysis, Cambridge University Press, Cambridge University Press, vol. 12(04), pages 627-627, November.
  10. Robert A. Jarrow, 2009. "The Term Structure of Interest Rates," Annual Review of Financial Economics, Annual Reviews, Annual Reviews, vol. 1(1), pages 69-96, November.
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Citations

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Cited by:
  1. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, Oxford University Press, edition 3, number 9780199574742, October.
  2. Ekeland, Ivar & Taflin, Erik, 2005. "A theory of bond portfolios," Economics Papers from University Paris Dauphine, Paris Dauphine University 123456789/6041, Paris Dauphine University.
  3. Antje Berndt & Peter Ritchken & Zhiqiang Sun, 2010. "On Correlation and Default Clustering in Credit Markets," Review of Financial Studies, Society for Financial Studies, Society for Financial Studies, vol. 23(7), pages 2680-2729, July.
  4. Carl Chiarella & Oh-Kang Kwon, 1999. "Forward Rate Dependent Markovian Transformations of the Heath-Jarrow-Morton Term Structure Model," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 5, Quantitative Finance Research Centre, University of Technology, Sydney.
  5. Carl Chiarella & Oh-Kang Kwon, 2001. "State Variables and the Affine Nature of Markovian HJM Term Structure Models," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 52, Quantitative Finance Research Centre, University of Technology, Sydney.
  6. Mikael Elhouar, 2008. "Finite-dimensional Realizations of Regime-switching HJM Models," Applied Mathematical Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 15(4), pages 331-354.
  7. Carl Chiarella & Samuel Chege Maina & Christina Nikitopoulos-Sklibosios, 2010. "Markovian Defaultable HJM Term Structure Models with Unspanned Stochastic Volatility," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 283, Quantitative Finance Research Centre, University of Technology, Sydney.
  8. Fred Espen Benth & Jukka Lempa, 2012. "Optimal portfolios in commodity futures markets," Papers 1204.2667, arXiv.org.
  9. Gapeev, Pavel V. & Küchler, Uwe, 2003. "On Markovian Short Rates in Term Structure Models Driven by Jump-Diffusion Processes," SFB 373 Discussion Papers 2003,44, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
  10. Björk, Tomas, 2000. "A Geometric View of Interest Rate Theory," Working Paper Series in Economics and Finance, Stockholm School of Economics 419, Stockholm School of Economics, revised 21 Dec 2000.
  11. Dai, Qiang & Singleton, Kenneth J., 2003. "Fixed-income pricing," Handbook of the Economics of Finance, Elsevier, in: G.M. Constantinides & M. Harris & R. M. Stulz (ed.), Handbook of the Economics of Finance, edition 1, volume 1, chapter 20, pages 1207-1246 Elsevier.
  12. Christina Nikitopoulos-Sklibosios, 2005. "A Class of Markovian Models for the Term Structure of Interest Rates Under Jump-Diffusions," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 6.
  13. Antje Berndt & Peter Ritchken & Zhiqiang Sun, . "On Correlation Effects and Default Clustering in Credit Models," GSIA Working Papers, Carnegie Mellon University, Tepper School of Business 2008-E36, Carnegie Mellon University, Tepper School of Business.
  14. Dorje C. Brody & Lane P. Hughston, 2011. "Interest Rates and Information Geometry," Papers 1111.3757, arXiv.org.
  15. Carl Chiarella & Sara Pasquali & Wolfgang Runggaldier, 2001. "On Filtering in Markovian Term Structure Models (An Approximation Approach)," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 65, Quantitative Finance Research Centre, University of Technology, Sydney.
  16. Björk, Tomas & Blix, Magnus & Landen, Camilla, 2005. "On finite dimensional realizations for the term structure of futures prices," Working Paper Series in Economics and Finance, Stockholm School of Economics 620, Stockholm School of Economics.

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