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On the Geometry of Interest Rate Models

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Author Info
Björk, Tomas () (Dept. of Finance, Stockholm School of Economics)
Abstract

In this paper, which is a substantial extension of the earlier essay Björk (2001), we give an overview of some recent work on the geometric 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 Markovian finite dimensional state space model.

We consider interest rate models of Heath-Jarrow-Morton type, where the forward rates are driven by a multidimensional Wiener process, and where he volatility is allowed to be an arbitrary smooth functional of the present forward rate curve. Within this framework 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. We also study stochastic volatility HJM models, and we provide a systematic method for the construction of concrete realizations.

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Publisher Info
Paper provided by Stockholm School of Economics in its series Working Paper Series in Economics and Finance with number 545.

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Length: 87 pages
Date of creation: 24 Nov 2003
Date of revision:
Handle: RePEc:hhs:hastef:0545

Note: To apppear in "Springer Lecture Notes in Mathematics"
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Related research
Keywords: Forward rate curves; interest rate models; factor models; state space models; Markovian realizations;

Other versions of this item:

Find related papers by JEL classification:
E43 - Macroeconomics and Monetary Economics - - Money and Interest Rates - - - Determination of Interest Rates; Term Structure of Interest Rates
G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

This paper has been announced in the following NEP Reports:

References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. Heath, David & Jarrow, Robert & Morton, Andrew, 1992. "Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation," Econometrica, Econometric Society, vol. 60(1), pages 77-105, January. [Downloadable!] (restricted)
  2. Ram Bhar & Carl Chiarella, 1995. "Transformation of Heath-Jarrow-Morton Models to Markovian Systems," Working Paper Series 53, School of Finance and Economics, University of Technology, Sydney. [Downloadable!]
  3. Tomas BjÃrk & Andrea Gombani, 1999. "Minimal realizations of interest rate models," Finance and Stochastics, Springer, vol. 3(4), pages 413-432. [Downloadable!] (restricted)
  4. 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. [Downloadable!] (restricted)
  5. 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. [Downloadable!] (restricted)
    Other versions:
  6. Hull, John & White, Alan, 1990. "Pricing Interest-Rate-Derivative Securities," Review of Financial Studies, Oxford University Press for Society for Financial Studies, vol. 3(4), pages 573-92. [Downloadable!] (restricted)
  7. Bent Jesper Christensen & Tomas Björk, . "Interest Rate Dynamics and Consistent Forward Rate Curves," Management Working Papers 1999-4, School of Economics and Management, University of Aarhus. [Downloadable!]
    Other versions:
  8. 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. [Downloadable!] (restricted)
  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. [Downloadable!]
  10. 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. [Downloadable!] (restricted)
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