LIBOR Market Model

From Wikipedia, the free encyclopedia

1 Model
- 1.1 Model Dynamic
2 Literature
- 2.1 Original Articles
- 2.2 Recommended Books
3 External links

[edit] 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.

[edit] Model Dynamic

The LIBOR market model models a set of $n$ forward rates $L i$ , $i=1,\ldots,n$ as lognormal processes

$\frac{d L_{i}(t)}{L_{i}(t)} = \mu_{i}(t) \ d t \ + \ \sigma_{i}(t) \ d W_{i} \text{,} \qquad i=1,\ldots,n \text{.}$

Here, $L i$ denotes the forward rate for the period $[T i, T i + 1]$ . 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.

[edit] Literature

[edit] 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.

[edit] 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.