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A deterministic-shift extension of analytically-tractable and time-homogeneous short-rate models

Author

Listed:
  • Damiano Brigo

    (Product and Business Development Group, Banca IMI, SanPaolo IMI Group, Corso Matteotti 6, 20121 Milano, Italy Manuscript)

  • Fabio Mercurio

    (Product and Business Development Group, Banca IMI, SanPaolo IMI Group, Corso Matteotti 6, 20121 Milano, Italy Manuscript)

Abstract

In the present paper we show how to extend any time-homogeneous short-rate model to a model that can reproduce any observed yield curve, through a procedure that preserves the possible analytical tractability of the original model. In the case of the Vasicek (1977) model, our extension is equivalent to that of Hull and White (1990), whereas in the case of the Cox-Ingersoll-Ross (1985) (CIR) model, our extension is more analytically tractable and avoids problems concerning the use of numerical solutions. We also consider the extension of time-homogeneous models without analytical formulas. We then explain why the CIR model is the most interesting model to be extended through our procedure, analyzing it in detail. We also consider an example of calibration to the cap market for two of the presented models. We finally hint at the same extension for multifactor models and explain its advantages for applications.

Suggested Citation

  • Damiano Brigo & Fabio Mercurio, 2001. "A deterministic-shift extension of analytically-tractable and time-homogeneous short-rate models," Finance and Stochastics, Springer, vol. 5(3), pages 369-387.
    • Handle: RePEc:spr:finsto:v:5:y:2001:i:3:p:369-387
    Note: received: October 1998; final version received: August 2000
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    Citations

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    Cited by:

    1. Breton, Michèle & Marzouk, Oussama, 2018. "Evaluation of counterparty risk for derivatives with early-exercise features," Journal of Economic Dynamics and Control, Elsevier, vol. 88(C), pages 1-20.
    2. Shane Miller, 2007. "Pricing of Contingent Claims Under the Real-World Measure," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 2-2007, January-A.
    3. Christa Cuchiero & Claudio Fontana & Alessandro Gnoatto, 2016. "A general HJM framework for multiple yield curve modelling," Finance and Stochastics, Springer, vol. 20(2), pages 267-320, April.
    4. Marco Di Francesco & Kevin Kamm, 2021. "How to handle negative interest rates in a CIR framework," Papers 2106.03716, arXiv.org.
    5. Keiichi Tanaka & Takeshi Yamada & Toshiaki Watanabe, 2010. "Applications of Gram-Charlier expansion and bond moments for pricing of interest rates and credit risk," Quantitative Finance, Taylor & Francis Journals, vol. 10(6), pages 645-662.
    6. Renne, Jean-Paul, 2016. "A tractable interest rate model with explicit monetary policy rates," European Journal of Operational Research, Elsevier, vol. 251(3), pages 873-887.
    7. Damiano Brigo & Mirela Predescu & Agostino Capponi, 2010. "Credit Default Swaps Liquidity modeling: A survey," Papers 1003.0889, arXiv.org, revised Mar 2010.
    8. Christa Cuchiero & Claudio Fontana & Alessandro Gnoatto, 2019. "Affine multiple yield curve models," Mathematical Finance, Wiley Blackwell, vol. 29(2), pages 568-611, April.
    9. Oh Kang Kwon, 2007. "Mean Reversion Level Extensions of Time-Homogeneous Affine Term Structure Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(4), pages 291-302.
    10. Giuseppe Orlando & Michele Bufalo, 2021. "Interest rates forecasting: Between Hull and White and the CIR#—How to make a single‐factor model work," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 40(8), pages 1566-1580, December.
    11. Shane Miller, 2007. "Pricing of Contingent Claims Under the Real-World Measure," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 25, July-Dece.
    12. Antonio Mannolini & Carlo Mari & Roberto Renò, 2008. "Pricing caps and floors with the extended CIR model," International Journal of Finance & Economics, John Wiley & Sons, Ltd., vol. 13(4), pages 386-400.
    13. Shane Miller & Eckhard Platen, 2004. "A Two-Factor Model for Low Interest Rate Regimes," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 11(1), pages 107-133, March.
    14. Hans-Peter Bermin, 2012. "Bonds and Options in Exponentially Affine Bond Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 19(6), pages 513-534, December.
    15. Claudio Fontana & Alessandro Gnoatto & Guillaume Szulda, 2019. "Multiple yield curve modelling with CBI processes," Papers 1911.02906, arXiv.org, revised Oct 2020.
    16. Claudio Fontana, 2022. "Caplet pricing in affine models for alternative risk-free rates," Papers 2202.09116, arXiv.org, revised Jan 2023.
    17. Pacati, Claudio & Pompa, Gabriele & Renò, Roberto, 2018. "Smiling twice: The Heston++ model," Journal of Banking & Finance, Elsevier, vol. 96(C), pages 185-206.
    18. Morelli, Giacomo & Santucci de Magistris, Paolo, 2019. "Volatility tail risk under fractionality," Journal of Banking & Finance, Elsevier, vol. 108(C).
    19. Cousin, Areski & Jiao, Ying & Robert, Christian Y. & Zerbib, Olivier David, 2016. "Asset allocation strategies in the presence of liability constraints," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 327-338.
    20. Oh Kwon, 2009. "On the equivalence of a class of affine term structure models," Annals of Finance, Springer, vol. 5(2), pages 263-279, March.
    21. Giuseppe Orlando & Rosa Maria Mininni & Michele Bufalo, 2018. "On The Calibration of Short-Term Interest Rates Through a CIR Model," Papers 1806.03683, arXiv.org.
    22. Markus Hess, 2020. "A pure-jump mean-reverting short rate model," Papers 2006.14814, arXiv.org.
    23. Martino Grasselli & Giulio Miglietta, 2016. "A flexible spot multiple-curve model," Quantitative Finance, Taylor & Francis Journals, vol. 16(10), pages 1465-1477, October.
    24. Yue Zhou, 2020. "Rational Kernel on Pricing Models of Inflation Derivatives," Papers 2001.05124, arXiv.org, revised Jan 2020.

    More about this item

    Keywords

    Short-rate models; Analytical tractability; Exponential Vasicek model; Cox-Ingersoll-Ross' model; Calibration to market data;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

    Lists

    This item is featured on the following reading lists, Wikipedia, or ReplicationWiki pages:
    1. Cox–Ingersoll–Ross model in Wikipedia English
    2. مدل کاکس-اینگرسول-راس in Wikipedia Persian

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