Refer to Brigo Ch.6.
The 'calibration' of LMM to market data is trivial - since LMM is designed so as to be consistent with Black volatilities, the 'calibration' amounts to simply bootstrapping the caplet vol. from the cap vol.
However, we don't just want to use LMM to price caps/floors. We want to price other more exotic products. Hence we also have to specify the instantaneous volatilities (the diffusion coefficient sigma in the SDE) and the instantaneous correlations (the rho among the Brownian motions). Remember, if the payoff of an instrument depends on more than one LIBOR forward rates, then it depends on the terminal correlation, which in turn depends on both instantaneous volatility AND instantaneous correlation(Brigo pp.234). Note that we are assuming the scalar LIBOR specification (see here).
# LMM requires that sigma be deterministic. Hence parameterizing sigma amounts to choosing a deterministic function (of time) for the volatility term structure. Parameterizing sigma has nothing to do with fitting to market data.
# Meanwhile, correlation is a constant (matrix). By parameterizing the rho matrix we reduce the rank (i.e. degree of freedom) of the matrix. Parameterizing rho has nothing to do with fitting to market data.
Update:
ReplyDeleteIn practice, suppose we choose to work under the LMM framework we still want to fit the model so that it prices swaptions correctly. Hence the parameterization of volatilities is in fact not totally free, but constrained by the swaption prices. The parameterization of correlation is more tricky (see Brigo pp.287-291). Basically we can require rho to be either exogenous or endogenous.