The LIBOR Market Model

The origins of the LIBOR market model can be traced to the practically motivated applications of Black's [1976] option formula for pricing derivatives in the LIBOR-based interest rate derivatives market. By using the LIBOR rate as the underlying asset and the current forward rate as the expectation of the future LIBOR rate, and assuming constant volatility, the Black formula immediately gives prices for European options written on the future LIBOR rate. Initially, the theoretical underpinnings of this approach seemed dubious. The main insight behind the development of the LMM is that the discretely compounded LIBOR rate can be represented as a portfolio of two zero-coupon bonds, and hence, it is a traded asset. By the virtue of being a traded asset, the method of martingale valuation could be used for pricing options based on the LIBOR rate. Since the expectation of the LIBOR rate can be shown to be the current forward rate under the chosen “forward measure,” the Black-type formulas are justified for pricing options. Further, since the Black implied volatilities can be given as different constants for different option expiration dates, these are consistent with a time-dependent volatility function for the term structure of volatilities of forward rate changes. This chapter illustrates a basic description of the LMM, arguably the most widely used preference-free term structure model for pricing fixed income derivatives.