LIBOR market 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, ratchet caps and floors, target redemption notes, autocaps, zero coupon swaptions, constant maturity swaps and spread options, among many others. The quantities that are modeled, rather than the short rate or instantaneous forward rates (like in the Heath-Jarrow-Morton framework) are a set of forward rates (also called LIBORs), which have the advantage of being directly observable in the market, and whose volatilities are naturally linked to traded contracts. Each forward rate is modeled by a lognormal process under its forward measure, i.e. a Black model leading to a Black formula for interest rate caps. This formula is the market standard to quote cap prices in terms of implied volatilities, hence the term "market model". The LIBOR market model may be interpreted as a collection of forward LIBOR dynamics for different forward rates with spanning tenors and maturities, each forward rate being consistent with a Black interest rate caplet formula for its canonical maturity. One can write the different rates dynamics under a common pricing measure, for example the forward measure for a preferred single maturity, and in this case forward rates will not be lognormal under the unique measure in general, leading to the need of numerical methods such as monte carlo simulation or approximations like the frozen drift assumption.

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.
• 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.

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.
• Damiano Brigo's lecture notes on the LIBOR market model for the Bocconi University fixed income course
• Market model