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Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach by Hong Xie, Chaoqun Ma, Guojun Gan

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CHAPTER 47

LIBOR MARKET MODELS

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.

47.1 Basic Concepts and Facts

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

equation

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

equation

where β(·) is defined in Definition 45.3.

For tT, 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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