Libor Market Models

Silke Prohl
Princeton University; NYU

April 30, 2012

Abstract:

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.

Libor Market Models

Interest Rate Markets

Let us consider a discrete tenor structure Screenshot_8, with constant tenor length Screenshot_9. The following notation is introduced for convenience; Screenshot_10. Let us denote by B(t, T) the time-t price of a zero coupon bond maturing at time Screenshot_11 the time-t forward rate settled at time T and received at time Screenshot_14 and by Screenshot_27 the time-t forward price associated to the dates T and U. These fundamental quantities are related by the following basic equation:

Screenshot_15

Throughout this work, Screenshot_16 denotes a complete stochastic basis, where Screenshot_17, and Screenshot_18. one can denote by Screenshot_19 the class of martingales on Screenshot_20 with respect to the measure Screenshot_21. 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

Screenshot_23

while they are related to the terminal forward measure via

Screenshot_24

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: Screenshot_25.
  • (A2T)he model should be arbitrage-free: Screenshot_26.
  • (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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