# LIBOR Market Model

LIBOR Market Model

= Model =

The LIBOR Market Model, also known as the BGM Model (Brace Gatarek Musiela Model, in reference of the names of some of the inventors), is a financial model of interest rates. It is used for pricing interest rate derivatives, especially exotic derivatives like Bermudan swaptions. The model primitives are a set of forward rates (also called LIBORs), which have the advantage of being directly observable in the market. Each forward rate is modeled by a lognormal process, i.e. a Black model. Thus the LIBOR market model may be interpreted as a collection of Black models considered under a common pricing measure.

Model Dynamic

The LIBOR market model models a set of $n$ forward rates $L_\left\{i\right\}$, $i=1,ldots,n$ as lognormal processes:$frac\left\{d L_\left\{i\right\}\left(t\right)\right\}\left\{L_\left\{i\right\}\left(t\right)\right\} = mu_\left\{i\right\}\left(t\right) d t + sigma_\left\{i\right\}\left(t\right) d W_\left\{i\right\} ext\left\{,\right\} qquad i=1,ldots,n ext\left\{.\right\}$Here, $L_\left\{i\right\}$ denotes the forward rate for the period $\left[T_\left\{i\right\},T_\left\{i+1\right\}\right]$. For each single forward rate the model corresponds to the Black model. The novelty is that, in contrast to the Black model, the LIBOR market model describes the dynamic of a whole family of forward rates under a common measure.

Literature

Original Articles

* Alan Brace, Dariusz Gatarek, Marek Musiela: The Market Model of Interest Rate Dynamics. Mathematical Finance 7, page 127. Blackwell 1997.
* Farshid Jamshidian: LIBOR and Swap Market Models and Measures, Finance and Stochastics 1, 1997, 293-330.
* Kristian R. Miltersen, Klaus Sandmann, Dieter Sondermann: Closed Form Solutions for Term Structure Derivatives with Lognormal Interest Rates. Journal of Finance 52, 409-430. 1997.

Recommended Books

* Alan Brace: "Engineering BGM". Chapman & Hall, 2008. ISBN 1-584-88968-3.
* Damiano Brigo, Fabio Mercurio: "Interest Rate Models - Theory and Practice". Springer, Berlin, 2001. ISBN 3-540-41772-9.
* Christian P. Fries: "Mathematical Finance: Theory, Modeling, Implementation". Wiley, 2007. ISBN 0470047224.
* Dariusz Gatarek, Przemyslaw Bachert, Robert Maksymiuk: "The LIBOR Market Model in Practice". John Wiley & Sons, 2007. ISBN 0-470-01443-1.
* Marek Musiela, Marek Rutkowski: "Martingale Methods in Financial Modelling: Theory and Applications". Springer, 1997. ISBN 3-540-61477-X.
* Riccardo Rebonato: "Modern Pricing of Interest-Rate Derivatives: The Libor Market Model and Beyond". Princeton University Press, 2002. ISBN 0-691-08973-6.
* John Schoenmakers: "Robust Libor Modelling and Pricing of Derivative Products". Chapman & Hall, 2004. ISBN 1-584-88441-X.

* [http://www.christian-fries.de/finmath/applets/ Java applets for pricing under a LIBOR market model and Monte-Carlo methods]
* [http://www.christian-fries.de/finmath/book/ Sample chapters of the book "Mathematical Finance" (ISBN 0470047224)] , with, e.g,, a derivation of the LIBOR market model drift.

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