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
Financial modeling is the task of building an abstraction, abstract representation (a mathematical model, model) of a real world finance, financial situation. This is a mathematical model designed to represent (a simplified version of) the perfor ...
of
interest rate
An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...
s. It is used for pricing
interest rate derivative
In finance, an interest rate derivative (IRD) is a derivative whose payments are determined through calculation techniques where the underlying benchmark product is an interest rate, or set of different interest rates. There are a multitude of diff ...
s, 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 The Heath–Jarrow–Morton (HJM) framework is a general framework to model the evolution of interest rate curves – instantaneous forward rate curves in particular (as opposed to simple forward rates). When the volatility and drift of the in ...
) are a set of
forward rates (also called forward
LIBOR
The London Inter-Bank Offered Rate is an interest-rate average calculated from estimates submitted by the leading banks in London. Each bank estimates what it would be charged were it to borrow from other banks. The resulting average rate is u ...
s), 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
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
process under its
forward measure
Forward is a relative direction, the opposite of backward.
Forward may also refer to:
People
* Forward (surname)
Sports
* Forward (association football)
* Forward (basketball), including:
** Point forward
** Power forward (basketball)
** Sma ...
, i.e. a
Black model
The Black model (sometimes known as the Black-76 model) is a variant of the Black–Scholes option pricing model. Its primary applications are for pricing options on future contracts, bond options, interest rate cap and floors, and swaptions. It w ...
leading to a Black formula for
interest rate cap An interest rate cap is a type of interest rate derivative in which the buyer receives payments at the end of each period in which the interest rate exceeds the agreed strike price. An example of a cap would be an agreement to receive a payment for ...
s. 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
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
, for example the
forward measure
Forward is a relative direction, the opposite of backward.
Forward may also refer to:
People
* Forward (surname)
Sports
* Forward (association football)
* Forward (basketball), including:
** Point forward
** Power forward (basketball)
** Sma ...
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
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...
or approximations like the frozen drift assumption.
Model dynamic
The LIBOR market models a set of
forward rates
,
as
lognormal processes. Under the respective
-Forward measure
:
Here we can consider that
(centered process).
Here,
is the forward rate for the period