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Phenomenology of the interest rate curve

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  • Jean-Philippe Bouchaud

    (Science & Finance, Capital Fund Management
    CEA Saclay;)

  • Nicolas Sagna
  • Rama Cont

    (Science & Finance, Capital Fund Management)

  • Nicole El-Karoui
  • Marc Potters

    (Science & Finance, Capital Fund Management)

Abstract

This paper contains a phenomenological description of the whole U.S. forward rate curve (FRC), based on an data in the period 1990-1996. We find that the average FRC (measured from the spot rate) grows as the square-root of the maturity, with a prefactor which is comparable to the spot rate volatility. This suggests that forward rate market prices include a risk premium, comparable to the probable changes of the spot rate between now and maturity, which can be understood as a `Value-at-Risk' type of pricing. The instantaneous FRC however departs form a simple square-root law. The distortion is maximum around one year, and reflects the market anticipation of a local trend on the spot rate. This anticipated trend is shown to be calibrated on the past behaviour of the spot itself. We show that this is consistent with the volatility `hump' around one year found by several authors (and which we confirm). Finally, the number of independent components needed to interpret most of the FRC fluctuations is found to be small. We rationalize this by showing that the dynamical evolution of the FRC contains a stabilizing second derivative (line tension) term, which tends to suppress short scale distortions of the FRC. This shape dependent term could lead, in principle, to arbitrage. However, this arbitrage cannot be implemented in practice because of transaction costs. We suggest that the presence of transaction costs (or other market `imperfections') is crucial for model building, for a much wider class of models becomes eligible to represent reality.

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Bibliographic Info

Paper provided by Science & Finance, Capital Fund Management in its series Science & Finance (CFM) working paper archive with number 500048.

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Date of creation: Dec 1997
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Publication status: Published in Applied Mathematical Finance, 6, 209, (1999)
Handle: RePEc:sfi:sfiwpa:500048

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  1. Hull, John & White, Alan, 1993. "One-Factor Interest-Rate Models and the Valuation of Interest-Rate Derivative Securities," Journal of Financial and Quantitative Analysis, Cambridge University Press, Cambridge University Press, vol. 28(02), pages 235-254, June.
  2. Schloegl, Erik & Daniel Sommer, 1997. "Factor Models and the Shape of the Term Structure," Discussion Paper Serie B, University of Bonn, Germany 395, University of Bonn, Germany.
  3. Rendleman, Richard J, Jr & Carabini, Christopher E, 1979. "The Efficiency of the Treasury Bill Futures Market," Journal of Finance, American Finance Association, American Finance Association, vol. 34(4), pages 895-914, September.
  4. 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, Econometric Society, vol. 60(1), pages 77-105, January.
  5. Cox, John C. & Ingersoll, Jonathan Jr. & Ross, Stephen A., 1981. "The relation between forward prices and futures prices," Journal of Financial Economics, Elsevier, Elsevier, vol. 9(4), pages 321-346, December.
  6. 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.
  7. Ho, Thomas S Y & Lee, Sang-bin, 1986. " Term Structure Movements and Pricing Interest Rate Contingent Claims," Journal of Finance, American Finance Association, American Finance Association, vol. 41(5), pages 1011-29, December.
  8. Chan, K C, et al, 1992. " An Empirical Comparison of Alternative Models of the Short-Term Interest Rate," Journal of Finance, American Finance Association, American Finance Association, vol. 47(3), pages 1209-27, July.
  9. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, Elsevier, vol. 5(2), pages 177-188, November.
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Cited by:
  1. Belal E. Baaquie, 2001. "Quantum Field Theory of Forward Rates with Stochastic Volatility," Papers, arXiv.org cond-mat/0110506, arXiv.org.
  2. Baaquie, Belal E. & Yang, Cao, 2009. "Empirical analysis of quantum finance interest rates models," Physica A: Statistical Mechanics and its Applications, Elsevier, Elsevier, vol. 388(13), pages 2666-2681.
  3. Baaquie, Belal E. & Liang, Cui & Warachka, Mitch C., 2007. "Hedging LIBOR derivatives in a field theory model of interest rates," Physica A: Statistical Mechanics and its Applications, Elsevier, Elsevier, vol. 374(2), pages 730-748.
  4. Rama Cont, 2005. "Modeling Term Structure Dynamics: An Infinite Dimensional Approach," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., World Scientific Publishing Co. Pte. Ltd., vol. 8(03), pages 357-380.
  5. repec:sfi:sfiwpa:313238 is not listed on IDEAS
  6. Belal E. Baaquie, 1998. "Quantum Field Theory of Treasury Bonds," Papers, arXiv.org cond-mat/9809199, arXiv.org.
  7. Roncoroni, Andrea & Galluccio, Stefano & Guiotto, Paolo, 2010. "Shape factors and cross-sectional risk," Journal of Economic Dynamics and Control, Elsevier, Elsevier, vol. 34(11), pages 2320-2340, November.
  8. Markus Leippold & Liuren Wu, 2002. "Design and Estimation of Quadratic Term Structure Models," Finance, EconWPA 0207014, EconWPA.
  9. D. Sornette, 1998. "``String'' formulation of the Dynamics of the Forward Interest Rate Curve," Papers, arXiv.org cond-mat/9802136, arXiv.org.
  10. Rene Carmona & Michael Tehranchi, 2004. "A Characterization of Hedging Portfolios for Interest Rate Contingent Claims," Papers, arXiv.org math/0407119, arXiv.org.
  11. repec:sfi:sfiwpa:500064 is not listed on IDEAS
  12. Rama Cont, 1999. "Modeling interest rate dynamics: an infinite-dimensional approach," Papers, arXiv.org cond-mat/9902018, arXiv.org.

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