Thoughts/Summary on LMM

As compared to short rate models:

In short rate models, we are trying to infer derivative prices by considering the dynamics of the short rate (bottom-up). For LMM the approach is more like top-down: since the market has always been using the Black76' convention to quote cap and floor prices (volatilities), why not play along and jump on the bandwagon, pretend that the forward rate really behaves like a tradable asset and follows a GBM, and develop a set of forward rate dynamics that rules out arbitrage?

Notes:

1. The fact that forward rate F(t,T_1,T_2) is a martingale under the T_2 forward measure follows from the definition of forward rate F(t,T_1,T_2) = (p_1-p_2)/(tau p_2); while the fact that we demand F(t,T_1,T_2) follows a GBM (instead of, say, arithmetic BM) is to make it coincide with market quoting convention (Black76'). F(t,T_1,T_2) being martingale does not necessarily require F(t,T_1,T_2) follows GBM.
2. Bottom line: when using LMM we are not really thinking about forward rates evolving according to GBM; LMM is a way to be consistent with the market (in other words, preclude arbitrage given the Black pricing convention).