The LIBOR/SABR Market Models: A Critical Review

Abstract

This paper reviews the LIBOR market model (LMM) and the LMM-SABR model. While a plethora of interest rate models, such as fundamental models, single-plus models, double-plus models, and triple plus models, can be used for valuation of plain vanilla derivatives, only a few models such as the LMM and the LMM-SABR have been proposed as models that can hedge plain vanilla derivatives as well as value complex interest rate derivatives. However, given that LMM and LMM-SABR models are triple-plus models, they are calibrated to market prices by allowing time-inhomogeneous volatilities, and by changing numerous model inputs, period by period. Changing the model period by period and using time-inhomogeneous volatilities make risk-return analysis impossible under the physical measure. Further, this paper demonstrates that the LMM-SABR model is based on the highly questionable assumption of zero drifts for the volatility processes (under the forward rate specific measures), which has no economic justification, and can lead to explosive behavior for volatilities. We suggest high-dimensional affine and quadratic models that use fast analytical approximations (such as the fourier inversion method and the cumulant expansion method) for pricing caps and swaptions, as alternatives to the LMM and the LMM-SABR model.