Computational Management Science

, Volume 16, Issue 1–2, pp 275–295 | Cite as

Calibration of one-factor and two-factor Hull–White models using swaptions

  • Vincenzo Russo
  • Gabriele TorriEmail author
Original Paper


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.


One-factor Hull–White model Two-factor Hull–White model Calibration Swaption Coupon bond option 



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.


Vincenzo Russo and not his employer is solely responsible for any errors.


  1. Black F, Karasinski P (1991) Bond and option pricing when short rates are lognormal. Financ Anal J 47:52–59CrossRefGoogle Scholar
  2. Brigo D, Mercurio F (2006) Interest rate models: theory and practice, 2nd edn. Springer, BerlinGoogle Scholar
  3. Di Francesco, M (2012) A general Gaussian interest rate model consistent with the current term structure. ISRN Probab Stat 2012, Article ID 673607, 16 pagesGoogle Scholar
  4. Hull J, White A (1990) Pricing interest rate derivative securities. Rev Financ Stud 3:573–592CrossRefGoogle Scholar
  5. Hull J, White A (1994a) Numerical procedure for implementing term structure models I: single-factor models. J Deriv 2:7–16CrossRefGoogle Scholar
  6. Hull J, White A (1994b) Numerical procedure for implementing term structure models II: two factor models. J Deriv 2:37–47CrossRefGoogle Scholar
  7. Hull J, White A (2001) The general Hull–White model and supercalibration. Financ Anal J 57(6):34–43CrossRefGoogle Scholar
  8. Ingber L (1996) Adaptive simulated annealing (ASA): lessons learned. Control Cybern 25:33–54Google Scholar
  9. Jamshidian F (1989) An exact bond option formula. J Finance 44:205–209CrossRefGoogle Scholar
  10. Jamshidian F (1995) A simple class of square root interest rate models. Appl Math Finance 2:61–72CrossRefGoogle Scholar
  11. Longstaff FA, Schwartz ES (1992) Interest rate volatility and the term structure: a two-factor general equilibrium model. J Finance 47(4):1259–1282CrossRefGoogle Scholar
  12. Munk C (1999) Stochastic duration and fast coupon bond option pricing in multi-factor models. Rev Deriv Res 3(2):157–181CrossRefGoogle Scholar
  13. Pellser A (1996) A tractable interest rate model that guarantees positive interest rates. Rev Deriv Res 1:269–284CrossRefGoogle Scholar
  14. Russo V, Fabozzi FJ (2016a) A one-factor shifted squared gaussian term structure model for interest rate modeling. J Fixed Income 25:36–45CrossRefGoogle Scholar
  15. Russo V, Fabozzi FJ (2016b) Pricing coupon bond options and swaptions under the one-factor Hull-White model. J Fixed Income 25:76–82CrossRefGoogle Scholar
  16. Russo V, Fabozzi FJ (2017a) Calibrating short interest rate models in negative rate environments. J Deriv 24:80–92CrossRefGoogle Scholar
  17. Russo V, Fabozzi FJ (2017b) Pricing coupon bond options and swaptions under the two-factor Hull–White model. J Fixed Income 27:30–36CrossRefGoogle Scholar
  18. Schlenkrich S (2012) Efficient calibration of the Hull–White model. Optim Control Appl Methods 33(3):352–362CrossRefGoogle Scholar
  19. Schrager DF, Pelsser A (2006) Pricing swaptions and coupon bond options in affine term structure models? Math Finance 16:673–694CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Group Risk ManagementAssicurazioni Generali S.p.A.MilanItaly
  2. 2.Department of Management, Economics and Quantitative MethodsUniversity of BergamoBergamoItaly
  3. 3.Department of FinanceVŠB-TU OstravaOstravaCzech Republic

Personalised recommendations