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Modeling Term Structure Dynamics: An Infinite Dimensional Approach

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  • RAMA CONT

    (Centre de Mathématiques Appliquées, Ecole Polytechnique, F-91128 Palaiseau, France)

Abstract

Motivated by stylized statistical properties of interest rates, we propose a modeling approach in which the forward rate curve is described as a stochastic process in a space of curves. After decomposing the movements of the term structure into the variations of the short rate, the long rate and the deformation of the curve around its average shape, this deformation is described as the solution of a stochastic evolution equation in an infinite dimensional space of curves. In the case where deformations are local in maturity, this equation reduces to a stochastic PDE, of which we give the simplest example. We discuss the properties of the solutions and show that they capture in a parsimonious manner the essential features of yield curve dynamics: imperfect correlation between maturities, mean reversion of interest rates, the structure of principal components of forward rates and their variances. In particular we show that a flat, constant volatility structures already captures many of the observed properties. Finally, we discuss parameter estimation issues and show that the model parameters have a natural interpretation in terms of empirically observed quantities.

Suggested Citation

  • 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.
    • Handle: RePEc:wsi:ijtafx:v:08:y:2005:i:03:n:s0219024905003049
    DOI: 10.1142/S0219024905003049
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    References listed on IDEAS

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    1. Raphaël Douady, 2013. "Yield Curve Smoothing and Residual Variance of Fixed Income Positions," Post-Print hal-00666751, HAL.
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    Citations

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    Cited by:

    1. Rama Cont & Marvin S. Mueller, 2019. "A stochastic partial differential equation model for limit order book dynamics," Papers 1904.03058, arXiv.org, revised May 2021.
    2. Stefan Tappe, 2019. "Existence of affine realizations for stochastic partial differential equations driven by L\'evy processes," Papers 1907.00335, arXiv.org.
    3. Neeraj Bhauryal & Ana Bela Cruzeiro & Carlos Oliveira, 2024. "Pathwise Stochastic Control and a Class of Stochastic Partial Differential Equations," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1967-1990, November.
    4. Raphaël Douady & Zeyu Cao, 2020. "Sabr Type Stochastic Volatility Operator In Hilbert Space," Working Papers hal-03018478, HAL.
    5. Rama Cont & Marvin Muller, 2019. "A Stochastic Pde Model For Limit Order Book Dynamics," Working Papers hal-02090449, HAL.
    6. A. Brace & G. Fabbri & B. Goldys, 2007. "An Hilbert space approach for a class of arbitrage free implied volatilities models," Papers 0712.1343, arXiv.org, revised Dec 2007.
    7. Toshiyuki Nakayama & Stefan Tappe, 2022. "Distance between closed sets and the solutions to stochastic partial differential equations," Papers 2205.00279, arXiv.org, revised Oct 2024.
    8. Zhongliang Tuo, 2013. "Hedging Against the Interest-rate Risk by Measuring the Yield-curve Movement," Papers 1312.6841, arXiv.org.
    9. Bueno-Guerrero, Alberto & Moreno, Manuel & Navas, Javier F., 2015. "Stochastic string models with continuous semimartingales," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 433(C), pages 229-246.
    10. Sonja Cox & Sven Karbach & Asma Khedher, 2022. "An infinite‐dimensional affine stochastic volatility model," Mathematical Finance, Wiley Blackwell, vol. 32(3), pages 878-906, July.
    11. Jacek Jakubowski & Jerzy Zabczyk, 2007. "Exponential moments for HJM models with jumps," Finance and Stochastics, Springer, vol. 11(3), pages 429-445, July.
    12. Bibinger, Markus & Trabs, Mathias, 2020. "Volatility estimation for stochastic PDEs using high-frequency observations," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 3005-3052.
    13. Özkan Fehmi & Schmidt Thorsten, 2005. "Credit risk with infinite dimensional Lévy processes," Statistics & Risk Modeling, De Gruyter, vol. 23(4), pages 281-299, April.

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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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