Abstract
In this paper, we analize a novel approach for calibrating the one-factor and the two-factor Hull–White models using swaptions under a market-consistent framework. The technique is based on the pricing formulas for coupon bond options and swaptions proposed by Russo and Fabozzi (J Fixed Income 25:76–82, 2016b; J Fixed Income 27:30–36, 2017b). Under this approach, the volatility of the coupon bond is derived as a function of the stochastic durations. Consequently, the price of coupon bond options and swaptions can be calculated by simply applying standard no-arbitrage pricing theory given the equivalence between the price of a coupon bond option and the price of the corresponding swaption. This approach can be adopted to calibrate parameters of the one-factor and the two-factor Hull–White models using swaptions quoted in the market. It represents an alternative with respect to the existing approaches proposed in the literature and currently used by practitioners. Numerical analyses are provided in order to highlight the quality of the calibration results in comparison with existing models, addressing some computational issues related to the optimization model. In particular, calibration results and sensitivities are provided for the one- and the two-factor models using market data from 2011 to 2016. Finally, an out-of-sample analysis is performed in order to test the ability of the model in fitting swaption prices different from those used in the calibration process.
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Notes
Longstaff and Schwartz (1992).
Preliminary analyses show that the resolution of the discretization do not influence significantly the calibration. Results are available upon request and are not reported for brevity.
This approach is implemented in Di Francesco (2012).
See Russo and Fabozzi (2017a) for further details about models calibration practices under negative rates.
References
Black F, Karasinski P (1991) Bond and option pricing when short rates are lognormal. Financ Anal J 47:52–59
Brigo D, Mercurio F (2006) Interest rate models: theory and practice, 2nd edn. Springer, Berlin
Di Francesco, M (2012) A general Gaussian interest rate model consistent with the current term structure. ISRN Probab Stat 2012, Article ID 673607, 16 pages
Hull J, White A (1990) Pricing interest rate derivative securities. Rev Financ Stud 3:573–592
Hull J, White A (1994a) Numerical procedure for implementing term structure models I: single-factor models. J Deriv 2:7–16
Hull J, White A (1994b) Numerical procedure for implementing term structure models II: two factor models. J Deriv 2:37–47
Hull J, White A (2001) The general Hull–White model and supercalibration. Financ Anal J 57(6):34–43
Ingber L (1996) Adaptive simulated annealing (ASA): lessons learned. Control Cybern 25:33–54
Jamshidian F (1989) An exact bond option formula. J Finance 44:205–209
Jamshidian F (1995) A simple class of square root interest rate models. Appl Math Finance 2:61–72
Longstaff FA, Schwartz ES (1992) Interest rate volatility and the term structure: a two-factor general equilibrium model. J Finance 47(4):1259–1282
Munk C (1999) Stochastic duration and fast coupon bond option pricing in multi-factor models. Rev Deriv Res 3(2):157–181
Pellser A (1996) A tractable interest rate model that guarantees positive interest rates. Rev Deriv Res 1:269–284
Russo V, Fabozzi FJ (2016a) A one-factor shifted squared gaussian term structure model for interest rate modeling. J Fixed Income 25:36–45
Russo V, Fabozzi FJ (2016b) Pricing coupon bond options and swaptions under the one-factor Hull-White model. J Fixed Income 25:76–82
Russo V, Fabozzi FJ (2017a) Calibrating short interest rate models in negative rate environments. J Deriv 24:80–92
Russo V, Fabozzi FJ (2017b) Pricing coupon bond options and swaptions under the two-factor Hull–White model. J Fixed Income 27:30–36
Schlenkrich S (2012) Efficient calibration of the Hull–White model. Optim Control Appl Methods 33(3):352–362
Schrager DF, Pelsser A (2006) Pricing swaptions and coupon bond options in affine term structure models? Math Finance 16:673–694
Acknowledgements
Gabriele Torri acknowledges the support of the Czech Science Foundation (GACR) under Project 15-23699S and SP2017/32, an SGS research project of VSB-TU Ostrava.
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Vincenzo Russo and not his employer is solely responsible for any errors.
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Russo, V., Torri, G. Calibration of one-factor and two-factor Hull–White models using swaptions. Comput Manag Sci 16, 275–295 (2019). https://doi.org/10.1007/s10287-018-0323-z
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DOI: https://doi.org/10.1007/s10287-018-0323-z