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Recently I have been reading a lot on the market models.

One thing that keeps escaping me - why is the Libor-market model (LMM) assumed to e free of aritrage in continuous time ?

To me this means that there must be a Numeraire so that all the bond-price-processes $P(t,T^*)$ with $T^* >t$ and $T^* \in [0,T]$ (with $T>T^*$ ) are martingales und the measure associated with that numeraire.

To state that the LMM is free of aritrage someone must have determines the relevant numeraire and done all the necessary checks. Is there a book or paper where those resuls can be found?

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well generally only the discrete bonds associated to the ends of the forward rates are modelled. to make these be martingales the drifts of the rates are chosen to make them driftless.

for an extension to all bonds, see http://ssrn.com/abstract=1461285

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Glasserman on pg 166 has a very accessible introduction to LMM under spot and forward measures

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