The LIBOR Market Model
The LIBOR Market Model — also known as Brace, Gatarek, Musiela model — is an interest rate model capable of reproducing the correlation structure of forward rates. One-factor models are unable to reproduce this structure and therefore cannot price accurately derivatives whose prices reflect these correlations. A typical example of such derivatives are swaps paying a non-linear function of the difference two swap rates for two different maturities.The model is constructed by using a family of LIBOR rates:
, where is LIBOR forward rate starting at and ending at , following
The SABR LIBOR-Market Model
An important flaw of the LMM is known as sticky volatilities: if the model is calibrated in a highly volatile market it assumes that this high volatility lasts forever, which leads to inaccurate results.The SABR LMM attempts to address this issue. In this model, each LIBOR rate is assumed to follow a log-normal dynamic having stochastic volatility:
