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A Geometric View of Interest Rate Theory

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  • Björk, Tomas

    ()
    (Department of Finance)

Abstract

The purpose of this essay is to give an overview of some recent workconcerning structural properties of the evolution of the forward rate curve in an arbitrage free bond market. The main problems to be discussed are as follows. 1. When is a given forward rate model consistent with a given family of forward rate curves? 2. When can the inherently infinite dimensional forward rate process be realized by means of a finite dimensional state space model. We consider interest rate models of Heath-Jarrow-Morton type, where the forward rates are driven by a multi- dimensional Wiener process, and where he volatility is allowed to be an arbitrary smooth functional of the present forward rate curve. Within this framwork we give necessary and sufficient conditions for consistency, as well as for the existence of a finite dimensional realization, in terms of the forward rate volatilities.

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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 419.

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Length: 39 pages
Date of creation: 20 Dec 2000
Date of revision: 21 Dec 2000
Publication status: Published in Option pricing, Interest Rates and Risk Management, Jouini, Elyes, Cvitanic, Jaksa, Musiela, Marek (eds.), 2001, chapter 7, pages 241-277, Cambridge University Press.
Handle: RePEc:hhs:hastef:0419

Note: To appear in "Handbook of Mathematical Finance". Cambridge University Press
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Postal: The Economic Research Institute, Stockholm School of Economics, P.O. Box 6501, 113 83 Stockholm, Sweden
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Web page: http://www.hhs.se/
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Related research

Keywords: interest rates; Markovian realizations; forward rates; invariant manifold;

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References

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  1. Ernst Eberlein & Sebastian Raible, 1999. "Term Structure Models Driven by General Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 9(1), pages 31-53.
  2. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
  3. Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1985. "A Theory of the Term Structure of Interest Rates," Econometrica, Econometric Society, vol. 53(2), pages 385-407, March.
  4. Nelson, Charles R & Siegel, Andrew F, 1987. "Parsimonious Modeling of Yield Curves," The Journal of Business, University of Chicago Press, vol. 60(4), pages 473-89, October.
  5. Björk, Tomas & Christensen, Bent Jesper, 1997. "Interest Rate Dynamics and Consistent Forward Rate Curves," Working Paper Series in Economics and Finance 209, Stockholm School of Economics.
  6. repec:aah:aarmng:1999-4 is not listed on IDEAS
  7. Ho, Thomas S Y & Lee, Sang-bin, 1986. " Term Structure Movements and Pricing Interest Rate Contingent Claims," Journal of Finance, American Finance Association, vol. 41(5), pages 1011-29, December.
  8. 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.
  9. Ram Bhar & Carl Chiarella, 1995. "Transformation of Heath-Jarrow-Morton Models to Markovian Systems," Working Paper Series 53, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
  10. Björk, Tomas & Svensson, Lars, 1999. "On the Existence of Finite Dimensional Realizations for Nonlinear Forward Rate Models," Working Paper Series in Economics and Finance 338, Stockholm School of Economics.
  11. Robert A. Jarrow, 2009. "The Term Structure of Interest Rates," Annual Review of Financial Economics, Annual Reviews, vol. 1(1), pages 69-96, November.
  12. Alan Brace & Marek Musiela, 1994. "A Multifactor Gauss Markov Implementation Of Heath, Jarrow, And Morton," Mathematical Finance, Wiley Blackwell, vol. 4(3), pages 259-283.
  13. Tomas BjÃrk & Andrea Gombani, 1999. "Minimal realizations of interest rate models," Finance and Stochastics, Springer, vol. 3(4), pages 413-432.
  14. Darrell Duffie & Rui Kan, 1996. "A Yield-Factor Model Of Interest Rates," Mathematical Finance, Wiley Blackwell, vol. 6(4), pages 379-406.
  15. Inui, Koji & Kijima, Masaaki, 1998. "A Markovian Framework in Multi-Factor Heath-Jarrow-Morton Models," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 33(03), pages 423-440, September.
  16. Hull, John & White, Alan, 1990. "Pricing Interest-Rate-Derivative Securities," Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 573-92.
  17. Peter Ritchken & L. Sankarasubramanian, 1995. "Volatility Structures Of Forward Rates And The Dynamics Of The Term Structure," Mathematical Finance, Wiley Blackwell, vol. 5(1), pages 55-72.
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Cited by:
  1. Francis X. Diebold & Canlin Li, 2003. "Forecasting the Term Structure of Government Bond Yields," NBER Working Papers 10048, National Bureau of Economic Research, Inc.
  2. Carl Chiarella & Sara Pasquali & Wolfgang Runggaldier, 2001. "On Filtering in Markovian Term Structure Models (An Approximation Approach)," Research Paper Series 65, Quantitative Finance Research Centre, University of Technology, Sydney.

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