Forward Rate

FORWARD RATE VOLATILITIES, SWAP RATE VOLATILITIES, AND THE IMPLEMENTATION OF THE LIBOR MARKET MODEL John Hull and Alan White Joseph L. Rotman School of Management University of Toronto August, 1999

Abstract This paper is concerned with the implementation of the LIBOR market model and its extensions. It develops and tests a simple analytic approximation for calculating the volatilities used by the market to price European swap options from the volatilities used by the market to price interest rate caps. The approximation is found to be very accurate for the range of market parameters normally encountered. It enables swap option volatility skews to be implied from cap volatility skews. It also allows the LIBOR market model to be easily calibrated to broker quotes on caps and European swap options so that a wide range of non-standard interest rate derivatives can be valued.

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FORWARD RATE VOLATILITIES, SWAP RATE VOLATILITIES, AND THE IMPLEMENTATION OF THE LIBOR MARKET MODEL The most popular over-the-counter interest rate options are interest rate caps/floors and European swap options. The standard market models for valuing these instruments are versions of Black’s (1976) model. This model was originally developed for valuing options on commodity futures, but has found many other applications in financial engineering. When Black’s model is used to value a caplet (one element of an interest rate cap), the underlying interest rate is assumed to be lognormal. When it is used to value European swap options, the underlying swap rate is assumed to be lognormal. Researchers such as Jamshidian (1997) have shown that the cap/floor market model and the European swap option market model are each internally consistent in the sense that they do not permit arbitrage opportunities. However, they are not exactly consistent with each other. In recent years Brace, Gatarek, and Musiela (1997), Jamshidian (1997), and Miltersen, Sandmann, and Sondermann (1997) have developed what...