# LIBOR market model

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Published: 2026-10-25T19:04:39.983Z

Updated: 2026-10-25T19:04:39.983Z

The LIBOR market model, also known as the BGM Model (Brace Gatarek Musiela Model, in reference to 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

## Key facts

## Overview

The LIBOR market model, also known as the BGM Model (Brace Gatarek Musiela Model, in reference to 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 forward 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 for numerical methods such as Monte Carlo simulation or approximations like the frozen drift assumption.

## Model dynamic

The LIBOR market models a set of

n

\{\\displaystyle n\}

forward rates

L

j

\{\\displaystyle L\_\{j\}\}

,

j         =         1         ,         …         ,         n

\{\\displaystyle j=1,\\ldots ,n\}

as  lognormal processes. Under the respective

T

j

\{\\displaystyle T\_\{j\}\}

-Forward measure

Q

T

j                 +                 1

\{\\displaystyle Q\_\{T\_\{j+1\}\}\}

d

## Literature

Brace, A., Gatarek, D. et Musiela, M. (1997): “The Market Model of Interest Rate Dynamics”, Mathematical Finance, 7(2), 127-154. Miltersen, K., Sandmann, K. et Sondermann, D., (1997): “Closed Form Solutions for Term Structure Derivates with Log-Normal Interest Rates”, Journal of Finance, 52(1), 409-430. Wernz, J. (2020): “Bank Management and Control”, Springer Nature, 85-88

## External links

Java applets for pricing under a LIBOR market model and Monte-Carlo methods Jave source code and spreadsheet of a LIBOR market model, including calibration to swaption and product valuation Damiano Brigo's lecture notes on the LIBOR market model for the Bocconi University fixed income course

## Sources

- LIBOR market model · Wikipedia: https://en.wikipedia.org/?curid=5205955

Source: Wikipedia contributors — https://en.wikipedia.org/?curid=5205955&oldid=1351992147

Licence: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/

Adaptation notice: Reformatted for XPay; the source may be abridged, restructured or translated. Consult the cited revision for the original.