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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.
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