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Approximate Closed-Form Solutions for Pricing Zero-Coupon Bonds in the Zero Lower Bound Framework

Author

Listed:
  • Jae-Yun Jun

    (LyRIDS, ECE Paris, Graduate School of Engineering, 10 Rue Sextius Michel, 75015 Paris, France)

  • Yves Rakotondratsimba

    (LyRIDS, ECE Paris, Graduate School of Engineering, 10 Rue Sextius Michel, 75015 Paris, France
    Deceased.)

Abstract

After the 2007 financial crisis, many central banks adopted policies to lower their interest rates; the dynamics of these rates cannot be captured using classical models. Recently, Meucci and Loregian proposed an approach to estimate nonnegative interest rates using the inverse-call transformation. Despite the fact that their work is distinguished from others in the literature by their consideration of practical aspects, some technical difficulties still remain, such as the lack of analytic expression for the zero-coupon bond (ZCB) price. In this work, we propose novel approximate closed-form solutions for the ZCB price in the zero lower bound (ZLB) framework, when the underlying shadow rate is assumed to follow the classical one-factor Vasicek model. Then, a filtering procedure is performed using the Unscented Kalman Filter (UKF) to estimate the unobservable state variable (the shadow rate), and the model calibration is proceeded by estimating the model parameters using the Particle Swarm Optimization (PSO) algorithm. Further, empirical illustrations are given and discussed using (as input data) the interest rates of the AAA-rated bonds compiled by the European Central Bank ranging from 6 September 2004 to 21 June 2012 (a period that concerns the ZLB framework). Our approximate closed-form solution is able to show a good match between the actual and estimated yield-rate values for short and medium time-to-maturity values, whereas, for long time-to-maturity values, it is able to estimate the trend of the yield rates.

Suggested Citation

  • Jae-Yun Jun & Yves Rakotondratsimba, 2024. "Approximate Closed-Form Solutions for Pricing Zero-Coupon Bonds in the Zero Lower Bound Framework," Mathematics, MDPI, vol. 12(17), pages 1-33, August.
  • Handle: RePEc:gam:jmathe:v:12:y:2024:i:17:p:2690-:d:1466771
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    References listed on IDEAS

    as
    1. Christensen, Jens H.E. & Diebold, Francis X. & Rudebusch, Glenn D., 2011. "The affine arbitrage-free class of Nelson-Siegel term structure models," Journal of Econometrics, Elsevier, vol. 164(1), pages 4-20, September.
    2. Leippold, Markus & Wu, Liuren, 2002. "Asset Pricing under the Quadratic Class," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 37(2), pages 271-295, June.
    3. Black, Fischer, 1995. "Interest Rates as Options," Journal of Finance, American Finance Association, vol. 50(5), pages 1371-1376, December.
    4. Jing Cynthia Wu & Fan Dora Xia, 2016. "Measuring the Macroeconomic Impact of Monetary Policy at the Zero Lower Bound," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 48(2-3), pages 253-291, March.
    5. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    6. Leo Krippner, 2011. "Modifying Gaussian term structure models when interest rates are near the zero lower bound," CAMA Working Papers 2011-36, Centre for Applied Macroeconomic Analysis, Crawford School of Public Policy, The Australian National University.
    7. Attilio Meucci & Angela Loregian, 2016. "Neither “Normal” nor “Lognormal”: Modeling Interest Rates across All Regimes," Financial Analysts Journal, Taylor & Francis Journals, vol. 72(3), pages 68-82, May.
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