The LIBOR Market Model (LMM) is an interest rate model based on evolving LIBOR market forward rates. It is also known as the Brace-Gatarek-Musiela (BGM) model, after the authors of one of the first papers where it was introduced. In contrast to models that evolve the instantaneous short rate (Hull-White, Black-Karasinski models) or instantaneous forward rates (Heath-Jarrow-Morton model), which are not directly observable in the market, the objects modeled using LMM are market-observable quantities (LIBOR forward rates). This makes LMM popular with market practitioners. Another feature that makes the LMM popular is that it is consistent with the market standard approach for pricing caps using Black’s formula.
The LMM can be used to price any instrument whose pay-off can be decomposed into a set of forward rates. It assumes that the evolution of each forward rate is lognormal. Each forward rate has a time dependent volatility and time dependent correlations with the other forward rates. After specifying these volatilities and correlations, the forward rates can be evolved using Monte Carlo simulation. Expectations of discounted coupons are then calculated in order to determine the fair value of an instrument. For more information on the LMM see Brigo and Mercurio, Joshi and Hull.
The standard lognormal LMM does not produce the market-observed caplet volatility smile/skew. To produce a skew, the LMM can be extended to incorporate a local volatility model, a stochastic volatility model, a jump diffusion model, or some combination of the above. The next section discusses the standard lognormal LMM in detail. A sub-section at the end discusses the local volatility extensions.
Lognormal LIBOR Market Model
Assume that there are forward rates that describe the pay-off of an interest rate derivative. The evolution of each forward rate is described by the stochastic differential equation:
= a standard Wiener process and
= the instantaneous correlation between forward rates and .
The instantaneous lognormal volatility and drift of forward rate are and , respectively. Note that the drift for forward rate can be calculated from the other forward rates and their instantaneous volatilities and correlations. Thus, the instantaneous volatilities and correlations completely describe how forward rates will evolve in the future. The first thing to do is to specify the instantaneous volatilities and correlations.
There are three main approaches for determining the instantaneous volatilities and correlations of forward rates:
- They can be obtained from analyzing historical data.
- They can be obtained by calibrating the model to current market prices of caplets and European swaptions.
- The user can explicitly specify volatilities and correlations based on how he or she believes they will evolve in the future.
A combination of these three approaches can also be used.
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