Research article

Extension of SABR Libor Market Model to handle negative interest rates

  • Received: 15 January 2020 Accepted: 09 March 2020 Published: 16 March 2020
  • JEL Codes: G12, E43

  • 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.

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

    Related Papers:

  • 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.


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    [1] Anderson L, Andreason J (2000) Volatility skews and extensions of the libor market model. Appl Math Financ 7: 1-32. doi: 10.1080/135048600450275
    [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. doi: 10.1111/1467-9965.00028
    [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. doi: 10.1016/j.amc.2014.05.017
    [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. doi: 10.1016/j.jjie.2019.01.001
    [11] Joshi M, Rebonato R (2003) A stochastic-volatility displaced-diffusion extension of the LIBOR market model. Quant Financ 3: 458-469. doi: 10.1088/1469-7688/3/6/305
    [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. doi: 10.1016/j.camwa.2017.11.024
    [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. doi: 10.3934/QFE.2019.1.22
    [17] Piterbarg V (2003) A stochastic volatility forward LIBOR model with a term structure of volatility smiles. Appl Math Financ 12: 147-185. doi: 10.1080/1350486042000297225
    [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.
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