Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Extension of SABR Libor Market Model to handle negative interest rates

Corporate Model Risk, Wells Fargo, 1753 Pinnacle Dr, Fl 6, Mc Lean, VA 22102, USA

Variations of Libor Market Model (LMM), including Constant Elasticity of Variance-LMM (CEV-LMM) and Stochastic Alpha-Beta-Rho LMM (SABR-LMM), have become popular for modeling interest rate term structure. Nevertheless, the limitation of applying CEV-/SABR-LMM to model negative interest rates still exists. In this paper, we adopt the approach of Free-Boundary SABR (FB-SABR), which is an extension based on standard SABR. The key idea of FB-SABR is to apply absolute value of forward rate $|F_t|$ in the rate dynamic $\mathrm{d} F_t = |F_t|^\beta \sigma_t \mathrm{d} W_{t}$, which naturally allows interest rates to across zero boundary. We focus on introducing FB-SABR into LMM to handle volatility smile under negative rates. This new model, FB-SABR-LMM, can be used to price interest rate instruments with negative strikes as well as to recover implied volatility surface.
  Article Metrics

Keywords Libor Market Model (LMM); SABR; SABR-LMM; Free Boundary SABR; negative rate

Citation: Jie Xiong, Geng Deng, Xindong Wang. Extension of SABR Libor Market Model to handle negative interest rates. Quantitative Finance and Economics, 2020, 4(1): 148-171. doi: 10.3934/QFE.2020007


  • 1. Anderson L, Andreason J (2000) Volatility skews and extensions of the libor market model. Appl Math Financ 7: 1-32.    
  • 2. Antonov A, Konikov M, Spector M (2015) The free boundary SABR: Natural extension to negative rates. Risk.
  • 3. Balland P, Tran Q (2013) SABR goes normal. Risk, 76-81.
  • 4. Brace A (1997) The market model of interest rate dynamics. Math Financ 7: 127-147.    
  • 5. Brigo D, Mercurio F (2006) Interest Rate Models-Theory and Practice, Springer, New York.
  • 6. Chesney M, Yor M, Jeanblanc M (2009) Mathematical Methods for Financial Markets, Springer, United Kingdom.
  • 7. Ferreiro A, García-Rodríguez J, López-Salas J, et al. (2014) SABR/LIBOR market models: Pricing and calibration for some interest rate derivatives. Appl Math Comput 242: 65-89.
  • 8. Hagan P, Kumar D, Lesniewski A, et al. (2002) Managing smile risk. Wilmott Mag, 84-108.
  • 9. Henry-Labordere P (2007) Unifying the BGM and SABR models: A short ride in hyperbolic geometry. SSRN. Available from: https://ssrn.com/abstract=877762 or http://dx.doi.org/10.2139/ssrn.877762.
  • 10. Honda Y, Inoue J (2019) The effectiveness of the negative interest rate policy in Japan: An early assessment. J Japanese Int Econ 52: 142-153.    
  • 11. Joshi M, Rebonato R (2003) A stochastic-volatility displaced-diffusion extension of the LIBOR market model. Quant Financ 3: 458-469.    
  • 12. Kienitz J (2015) Approximate and PDE solution to the boundary free SABR model-application to pricing and calibration. Working Paper.
  • 13. LeFloch F, Kennedy G (2013) Finite difference techniques for arbitrage free SABR. Working Paper.
  • 14. López-Salas J, Vázquez C (2018) PDE formulation of some SABR/LIBOR market models and its numerical solution with a sparse grid combination technique. Comput Math Applications 75: 1616-1634.    
  • 15. Morini M, Mercurio F (2007) No-arbitrage dynamics for a tractable SABR term structure LIBOR model. Bloomberg Portfolio Res Pap.
  • 16. Pedersen H, Swanson N (2019) A survey of dynamic Nelson-Siegel models, diffusion indexes, and big data methods for predicting interest rates. Quant Financ Econ 3: 22-45.    
  • 17. Piterbarg V (2003) A stochastic volatility forward LIBOR model with a term structure of volatility smiles. Appl Math Financ 12: 147-185.
  • 18. Rebonato R (2002) Modern pricing of interest-rate derivatives, Princeton University Press.
  • 19. Rebonato R (2007) A time-homogeneous, SABR-consistent extension of the LMM: calibration and numerical results. Risk.
  • 20. Rebonato R, McKay K, White R (2009) The SABR/LIBOR Market Mode: Pricing, Calibration and Hedging for Complex Interest-Rate Derivertives, Wiley, United Kingdom.
  • 21. Schoenmakers J, Coeffey B (2000) Stable implied calibration of a multi-factor LIBOR model via a semi-parametric correlation structure. WIAS Working Paper.
  • 22. Wu L, Zhang F (2006) LIBOR market model with stochastic volatility. J Ind Manage Optim 2: 199-227.
  • 23. Zhu J (2007) An extended LIBOR market model with nested stochastic volatility dynamics. Available at SSRN 955352.


Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved