Instantaneous forward rates are not so simple to estimate, as we have seen. One may want to model other rates, such as LIBOR, directly. There has been some effort in the years after the publication of HJM (Econometrica 60:77–105, 1992) in 1992 to develop arbitrage-free models of other than instantaneous, continuously compounded rates. The breakthrough came 1997, when the LIBOR market models were introduced by Miltersen et al. (J. Finance 52:409–430, 1997) and Brace et al. (Math. Finance 7(2):127–155, 1997) who succeeded in finding a HJM-type model inducing lognormal LIBOR rates. At the same time, Jamshidian (Finance Stoch. 1(4):290–330, 1997) developed a framework for arbitrage-free LIBOR and swap rate models not based on HJM. The principal idea of these approaches is to choose a different numeraire than the risk-free account (the latter does not even necessarily have to exist). Both approaches lead to Black’s formula for either caps (LIBOR models) or swaptions (swap rate models). Because of this they are usually referred to as “market models”.
KeywordsMarket Model Implied Volatility Swap Rate LIBOR Rate Market Quote
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