Extension of SABR Libor Market Model to handle negative interest rates

Jie Xiong; Geng Deng; Xindong Wang; Jie Xiong; Geng Deng; Xindong Wang

doi:10.3934/QFE.2020007

Quantitative Finance and Economics

2020, Volume 4, Issue 1: 148-171. doi: 10.3934/QFE.2020007

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Research article

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

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.
- 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[J]. Quantitative Finance and Economics, 2020, 4(1): 148-171. doi: 10.3934/QFE.2020007

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Abstract

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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