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There are two form of LIBOR Market Model that has a drift introduced. I would like to know in plain english explanation why do practitioners use these changes of measure. Are there any significance to the pricing of interest rate derivatives like Caps? I know that caplet is priced in the zero coupon bond associated with the measure that makes the LIBOR Model in the form of zero-drift. But why do we have the model with drifts?

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the point of the LMM is to evolve several different rates simultaneously. If you have rates $f_i$ from $t_i$ to $t_{i+1}$ and take a bond expiring at $t_j$ as numeraire then only the rate $f_{j-1}$ is driftless.

Typically $P_{t_0}$ is used as numeraire which makes all the rates have drift. It generally gives lower variance.

(see my book More mathematical finance for extensive discussion of drifts.)

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