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Yield Curve Smoothing and Residual Variance of Fixed Income Positions

Author

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  • Raphaël Douady

    () (CES - Centre d'économie de la Sorbonne - UP1 - Université Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

Abstract

We model the yield curve in any given country as an object lying in an infinite-dimensional Hilbert space, the evolution of which is driven by what is known as a cylindrical Brownian motion. We assume that volatilities and correlations do not depend on rates (which hence are Gaussian). We prove that a principal component analysis (PCA) can be made. These components are called eigenmodes or principal deformations of the yield curve in this space. We then proceed to provide the best approximation of the curve evolution by a Gaussian Heath-Jarrow-Morton model that has a given finite number of factors. Finally, we describe a method, based on finite elements, to compute the eigenmodes using historical interest rate data series and show how it can be used to compute approximate hedges which optimize a criterion depending on transaction costs and residual variance.

Suggested Citation

  • Raphaël Douady, 2013. "Yield Curve Smoothing and Residual Variance of Fixed Income Positions," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00666751, HAL.
  • Handle: RePEc:hal:cesptp:hal-00666751
    DOI: 10.1007/978-3-319-02069-3_10
    Note: View the original document on HAL open archive server: https://hal.archives-ouvertes.fr/hal-00666751
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    References listed on IDEAS

    as
    1. D. P. Kennedy, 1994. "The Term Structure Of Interest Rates As A Gaussian Random Field," Mathematical Finance, Wiley Blackwell, vol. 4(3), pages 247-258.
    2. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters,in: Theory Of Valuation, chapter 8, pages 229-288 World Scientific Publishing Co. Pte. Ltd..
    3. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters,in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305 World Scientific Publishing Co. Pte. Ltd..
    4. Schaefer, Stephen M. & Schwartz, Eduardo S., 1984. "A Two-Factor Model of the Term Structure: An Approximate Analytical Solution," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 19(04), pages 413-424, December.
    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 1011-1029, December.
    6. Farshid Jamshidian, 1993. "Option and Futures Evaluation With Deterministic Volatilities," Mathematical Finance, Wiley Blackwell, vol. 3(2), pages 149-159.
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    Cited by:

    1. Jean-Philippe Bouchaud & Nicolas Sagna & Rama Cont & Nicole El-Karoui & Marc Potters, 1999. "Phenomenology of the interest rate curve," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(3), pages 209-232.
    2. Jean-Philippe BOUCHAUD & Rama CONT & Nicole EL KAROUI & Marc POTTERS & Nicolas SAGNA, 1997. "Phenomenology of the interest curve," Finance 9712009, University Library of Munich, Germany.
    3. Rene Carmona & Michael Tehranchi, 2004. "A Characterization of Hedging Portfolios for Interest Rate Contingent Claims," Papers math/0407119, arXiv.org.
    4. repec:wsi:ijtafx:v:08:y:2005:i:03:n:s0219024905003049 is not listed on IDEAS

    More about this item

    Keywords

    Cylindrical Brownian motion; Term structure of interest rates; Yield curve; Heath-Jarrow-Morton model; Fixed-income models; Asymptotic arbitrage;

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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