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Calibrating the Black-Derman-Toy model: some theoretical results

Author

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  • Phelim Boyle
  • Ken Seng Tan
  • Weidong Tian

Abstract

The Black-Derman-Toy (BDT) model is a popular one-factor interest rate model that is widely used by practitioners. One of its advantages is that the model can be calibrated to both the current market term structure of interest rate and the current term structure of volatilities. The input term structure of volatility can be either the short term volatility or the yield volatility. Sandmann and Sondermann derived conditions for the calibration to be feasible when the conditional short rate volatility is used. In this paper conditions are investigated under which calibration to the yield volatility is feasible. Mathematical conditions for this to happen are derived. The restrictions in this case are more complicated than when the short rate volatilities are used since the calibration at each time step now involves the solution of two non-linear equations. The theoretical results are illustrated by showing numerically that in certain situations the calibration based on the yield volatility breaks down for apparently plausible inputs. In implementing the calibration from period n to period n + 1, the corresponding yield volatility has to lie within certain bounds. Under certain circumstances these bounds become very tight. For yield volatilities that violate these bounds, the computed short rates for the period (n, n + 1) either become negative or else explode and this feature corresponds to the economic intuition behind the breakdown.

Suggested Citation

  • Phelim Boyle & Ken Seng Tan & Weidong Tian, 2001. "Calibrating the Black-Derman-Toy model: some theoretical results," Applied Mathematical Finance, Taylor & Francis Journals, vol. 8(1), pages 27-48.
    • Handle: RePEc:taf:apmtfi:v:8:y:2001:i:1:p:27-48
    DOI: 10.1080/13504860110062049
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    References listed on IDEAS

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    Cited by:

    1. Radu Tunaru, 2015. "Model Risk in Financial Markets:From Financial Engineering to Risk Management," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 9524, August.
    2. Grzegorz Krzy.zanowski & Andr'es Sosa, 2020. "Performance analysis of Zero Black-Derman-Toy interest rate model in catastrophic events: COVID-19 case study," Papers 2007.00705, arXiv.org, revised Jul 2020.
    3. Dan Pirjol, 2015. "Hogan-Weintraub singularity and explosive behaviour in the Black-Derman-Toy model," Quantitative Finance, Taylor & Francis Journals, vol. 15(7), pages 1243-1257, July.

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