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Yield Curve Smoothing and Residual Variance of Fixed Income Positions

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  • Raphaël Douady

    (Riskdata - Financial Risk Management Software, CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 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 optimise a criterion depending on transaction costs and residual variance.

Suggested Citation

  • Raphaël Douady, 2014. "Yield Curve Smoothing and Residual Variance of Fixed Income Positions," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01151276, HAL.
    • Handle: RePEc:hal:cesptp:hal-01151276
    Note: View the original document on HAL open archive server: https://hal.science/hal-01151276
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    References listed on IDEAS

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    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.
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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. Raphaël Douady & Zeyu Cao, 2020. "Sabr Type Stochastic Volatility Operator In Hilbert Space," Working Papers hal-03018478, HAL.
    4. Rene Carmona & Michael Tehranchi, 2004. "A Characterization of Hedging Portfolios for Interest Rate Contingent Claims," Papers math/0407119, arXiv.org.
    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.

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    More about this item

    Keywords

    interest rate models; arbitrage pricing; infinite dimensional models; Martingale methods;
    All these keywords.

    JEL classification:

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

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