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Phenomenology of the interest rate curve

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Author Info

  • Jean-Philippe Bouchaud
  • Nicolas Sagna
  • Rama Cont
  • Nicole El-Karoui
  • Marc Potters

Abstract

The paper contains a phenomenological description of the whole US forward rate curve (FRC), based on data in the period 1990-1996. It is found that the average deviation of the FRC 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 from a simple square-root law. The deformation 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. It is shown that this is consistent with the volatility 'hump' around one year found by several authors (which is confirmed). Finally, the number of independent components needed to interpret most of the FRC fluctuations is found to be small. This is rationalized 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 to arbitrage. However, this arbitrage cannot be implemented in practice because of transaction costs. It is suggested 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.1

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Bibliographic Info

Article provided by Taylor & Francis Journals in its journal Applied Mathematical Finance.

Volume (Year): 6 (1999)
Issue (Month): 3 ()
Pages: 209-232

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Handle: RePEc:taf:apmtfi:v:6:y:1999:i:3:p:209-232

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Related research

Keywords: Forward Rate Curve; Spot Rate; Risk Premium; 'value-at-risk' Pricing; Volatility Hump; Deformation;

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References

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  1. Schloegl, Erik & Daniel Sommer, 1997. "Factor Models and the Shape of the Term Structure," Discussion Paper Serie B 395, University of Bonn, Germany.
  2. Robert A. Jarrow, 2009. "The Term Structure of Interest Rates," Annual Review of Financial Economics, Annual Reviews, vol. 1(1), pages 69-96, November.
  3. Cox, John C. & Ingersoll, Jonathan Jr. & Ross, Stephen A., 1981. "The relation between forward prices and futures prices," Journal of Financial Economics, Elsevier, vol. 9(4), pages 321-346, December.
  4. Chan, K C, et al, 1992. " An Empirical Comparison of Alternative Models of the Short-Term Interest Rate," Journal of Finance, American Finance Association, vol. 47(3), pages 1209-27, July.
  5. 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.
  6. 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, vol. 28(02), pages 235-254, June.
  7. Rendleman, Richard J, Jr & Carabini, Christopher E, 1979. "The Efficiency of the Treasury Bill Futures Market," Journal of Finance, American Finance Association, vol. 34(4), pages 895-914, September.
  8. 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.
  9. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
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Citations

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Cited by:
  1. Rama Cont, 1999. "Modeling interest rate dynamics: an infinite-dimensional approach," Papers cond-mat/9902018, arXiv.org.
  2. D. Sornette, 1998. "``String'' formulation of the Dynamics of the Forward Interest Rate Curve," Papers cond-mat/9802136, arXiv.org.
  3. Jean-Philippe Bouchaud, 2002. "An introduction to statistical finance," Science & Finance (CFM) working paper archive 313238, Science & Finance, Capital Fund Management.
  4. Roncoroni, Andrea & Galluccio, Stefano & Guiotto, Paolo, 2010. "Shape factors and cross-sectional risk," Journal of Economic Dynamics and Control, Elsevier, vol. 34(11), pages 2320-2340, November.
  5. 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., vol. 8(03), pages 357-380.
  6. Markus Leippold & Liuren Wu, 2002. "Design and Estimation of Quadratic Term Structure Models," Finance 0207014, EconWPA.
  7. Belal Baaquie & Jean-Philippe Bouchaud, 2004. ""Stiff" Field Theory of Interest Rates and Psychological Future Time," Science & Finance (CFM) working paper archive 500064, Science & Finance, Capital Fund Management.
  8. Belal E. Baaquie, 1998. "Quantum Field Theory of Treasury Bonds," Papers cond-mat/9809199, arXiv.org.
  9. 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, vol. 374(2), pages 730-748.
  10. Baaquie, Belal E. & Yang, Cao, 2009. "Empirical analysis of quantum finance interest rates models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(13), pages 2666-2681.
  11. Belal E. Baaquie, 2001. "Quantum Field Theory of Forward Rates with Stochastic Volatility," Papers cond-mat/0110506, arXiv.org.
  12. Rene Carmona & Michael Tehranchi, 2004. "A Characterization of Hedging Portfolios for Interest Rate Contingent Claims," Papers math/0407119, arXiv.org.

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