Risk Latte - The Allure of the Libor Market Model (LMM)

The Allure of the Libor Market Model (LMM)

Team Latte
January 19, 2012

Libor Market Models (LMM), also sometimes referred to as the Market Model of Interest Rates, has come to dominate the interest rate modeling space. History of the development of various mathematical models to capture the interest rate (the short rate) and the yield curve dynamics offer rich insights into how practitioners have viewed the complex world of interest rates. In this long chain of evolution the LMM is the latest development.

Many top banks now use LMM to price exotic interest rate derivatives and despite the existence and use of other powerful models, such as the mean revering models of CIR, the Black-Derman-Toy (BDT) model and the Heath-Jarrow-Morton (HJM) model, the Libor Market Model has risen to prominence.

For a more detailed analysis of the LMM, we would like to refer the reader to our CFE Learning Centre. Here wed like to talk about some key features of LMM that has made the model so attractive for practitioners.

The Libor Market Model (LMM) was introduced by A. Brace, D. Gatarek and M. Musiela, in their seminal paper The Market Model of Interest Rate Dynamics. Interested readers may also refer to Fixed Income Securities by Bruce Tuckman and Angel Serratfor an excellent explanation of this model.

Two key aspects of LMM which distinguishes it and thereby makes it preferable to use from other interest rate models are:

  1. Unlike other factor models of interest rates which work with abstract mathematical factors, in LMM the factors are the observable forward rates, which cumulatively explain the yield curve.

  2. Unlike other models of interest rates, in LMM we do not have to specify the drift function. The LMM model can be completely described by the volatilities and the correlation of the forward rates

Quants and traders who have to price and trade exotic interest rate derivatives when the prices of other liquid derivatives are given find the LMM quite useful because of three main reasons:

  1. First, since the factors of the LMM are forward rates themselves the model is able to match the initial values and the market term structures quite easily;

  2. Secondly, since the number of factors are large the volatilities and correlation of interest rates across the term structure are captured very well by this model;

  3. Finally, it is quite easy to calibrate the LMM. This is because the prices of the most liquid instruments in the market have a direct link with the volatility of the forward rates.

Of course, one must bear in mind that LMM is a math heavy model. To understand LMM in detail one needs to navigate through some very dense mathematics.


References:
  1. The Market Model of Interest Rate Dynamics, Alan Brace, DariuszGatarek and Marek Musiela, Mathematical Finance, 1997.
  2. Fixed Income Securities, Bruce Tuckman and Angel Serrat, John Wiley & Sons, 2012.


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