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Calibration of the Ueno’s Shadow Rate Model of Interest Rates

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  • Lenka Košútová

    (Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina, 842 48 Bratislava, Slovakia)

  • Beáta Stehlíková

    (Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina, 842 48 Bratislava, Slovakia)

Abstract

Shadow rate models of interest rates are based on the assumption that the interest rates are determined by an unobservable shadow rate. This idea dates back to Fischer Black, who understood the interest rate as an option that cannot become negative. Its possible zero values are consequences of negative values of the shadow rate. In recent years, however, the negative interest rates have become a reality. To capture this behavior, shadow rate models need to be adjusted. In this paper, we study Ueno’s model, which uses the Vasicek process for the shadow rate and adjusts its negative values when constructing the short rate. We derive the probability properties of the short rate in this model and apply the maximum likelihood estimation method to obtain the parameters from the real data. The other interest rates are—after a specification of the market price of risk—solutions to a parabolic partial differential equation. We solve the equation numerically and use the long-term rates to fit the market price of risk.

Suggested Citation

  • Lenka Košútová & Beáta Stehlíková, 2024. "Calibration of the Ueno’s Shadow Rate Model of Interest Rates," Mathematics, MDPI, vol. 12(22), pages 1-12, November.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:22:p:3564-:d:1521265
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    References listed on IDEAS

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    2. Black, Fischer, 1995. "Interest Rates as Options," Journal of Finance, American Finance Association, vol. 50(5), pages 1371-1376, December.
    3. Realdon, Marco, 2016. "Tests of non linear Gaussian term structure models," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 44(C), pages 128-147.
    4. Marcel A. Priebsch, 2023. "Computing Arbitrage-Free Yields in Multi-Factor Gaussian Shadow Rate Term Structure Models," Quarterly Journal of Finance (QJF), World Scientific Publishing Co. Pte. Ltd., vol. 13(04), pages 1-30, December.
    5. Vasicek, Oldrich Alfonso, 1977. "Abstract: An Equilibrium Characterization of the Term Structure," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 12(4), pages 627-627, November.
    6. Nelson, Charles R & Siegel, Andrew F, 1987. "Parsimonious Modeling of Yield Curves," The Journal of Business, University of Chicago Press, vol. 60(4), pages 473-489, October.
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