LIBOR market model

Model

The LIBOR Market Model, also referred to as the BGM Model in industry, is an interest rate model used for the pricing of interest rate derivatives, especially exotic derivatives like Bermudan swaptions. The model primitives are a set of LIBOR forward rates, 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 $L_{i}$, $i=1,\ldots ,n$ as lognormal processes

${\frac {dL_{i}(t)}{L_{i}(t)}}=\mu _{i}(t)\ dt\ +\ \sigma _{i}(t)\ dW_{i}{\text{,}}\qquad i=1,\ldots ,n{\text{.}}$

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.

Literature

Original Articles

• Alan Brace, Dariusz Gatarek, Marek Musiela: The Market Model of Interest Rate Dynamics. Mathematical Finance 7, page 127. Blackwell 1997.
• 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.

External links

• Java applets for pricing under a LIBOR market model and Monte-Carlo methods
• Sample chapters of the book "Mathematical Finance" (ISBN 0470047224), with, e.g,, a derivation of the LIBOR market model drift.