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.

中文翻译：

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扩展SABR Libor市场模型以处理负利率

Libor市场模型（LMM）的变体，包括方差-LMM的恒定弹性（CEV-LMM）和随机Alpha-Beta-Rho LMM（SABR-LMM），已成为建模利率期限结构的流行方法。然而，使用CEV- / SABR-LMM建模负利率的局限性仍然存在。在本文中，我们采用自由边界SABR（FB-SABR）的方法，它是基于标准SABR的扩展。FB-SABR的关键思想是在汇率动态$ \ mathrm {d} F_t = | F_t | ^ \ beta \ sigma_t \ mathrm {d} W_ {t} $中应用远期汇率$ | F_t | $的绝对值，自然可以使利率跨越零边界。我们专注于将FB-SABR引入LMM以应对负利率下的波动性微笑。这款新型号FB-SABR-LMM