The short rate models and the HJM models introduced in the previous chapters concentrate on unobservable rates, including short rates and instantaneous forward rates. Brace et al. (1997) introduced LIBOR (London interbank offered rate) market models, which model the observable LIBOR rates directly. In this chapter, we introduce the LIBOR model and relevant concepts.

**Definition 47.1** (LIBOR). Let δ > 0 and 0 ≤ *t* < *T.* The δ-period forward LIBOR for the future date *T* prevailing at time *t* is defined as

**Definition 47.2** (*T*-Forward Measure). Let *Q* be an equivalent martingale measure for the bond market. Let *T* > 0. Then the *T*-forward measure on (Ω, _{T}) is defined as

where β(·) is defined in Definition 45.3.

For *t* ≤ *T*, the conditional expectation of given _{t}, written as , is defined as

**Definition 47.3** (LIBOR Market Model). Let δ > 0 and ...

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