Extension of SABR Libor Market Model to handle negative interest rates

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.

Keywords

• libor market model (lmm)
• sabr
• sabr-lmm
• free boundary sabr
• negative rate