Libor Market Models
Princeton University; NYU
April 30, 2012
This manuscript reviews standard classes of Libor Market models and discusses their numerical approximation machinery. It gives introduction to non-defaultable, defaultable models, Levy-forced models and affine Libor models.
Libor Market Models – Introductory remarks
There are two natural starting points for modeling LIBOR rates: the rate itself and the forward price. Although they differ only by an additive and a multiplicative constant, cf. (1.2.1), the model dynamics are noticeably different, depending on whether the model is based on the LIBOR or the forward price. In addition, the consequences from the point of view of econometrics are also significant.
Modeling LIBOR rates directly, leads to positive rates and arbitrage-free dynamics, but the model is not analytically tractable. On the other hand, models for the forward price are analytically tractable, but rates can become negative. The only models that can respect all properties simultaneously are market models.
Interest Rate Markets
Let us consider a discrete tenor structure , with constant tenor length . The following notation is introduced for convenience; . Let us denote by B(t, T) the time-t price of a zero coupon bond maturing at time the time-t forward rate settled at time T and received at time and by the time-t forward price associated to the dates T and U. These fundamental quantities are related by the following basic equation:
Throughout this work, denotes a complete stochastic basis, where , and . one can denote by the class of martingales on with respect to the measure . one can associate to each date Tk in the tenor structure a forward martingale measure, denoted by [k], k 2. By the definition of forward measures, cf. [Def. 14.1.1]MusielaRutkowski97, the bond price with maturity Tk is the numeraire for the forward measure [k]. Thus, one can have that forward measures are related to each other via
while they are related to the terminal forward measure via
All forward measures are assumed to be equivalent to the measure aa.
Axioms for models
In this section, one can present and discuss certain requirements that a model for rates should satisfy. These requirements are motivated both by the economic and financial aspects of rates, as well as by the practical demands for implementing and using a model in practice. The aim here is to unify the line of thought in HuntKennedyPelsser00 and in KellerResselPapapantoleonTeichmann09 KellerResselPapapantoleonTeichmann09.
A model for rates should satisfy the following requirements:
- (A1r)ates should be non-negative: .
- (A2T)he model should be arbitrage-free: .
- (A3T)he model should be analytically tractable, easy to implement and quick to calibrate to market data.
- (A4T)he model should provide a good calibration to market data of liquid derivatives, i.e. caps and swaptions.
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