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Barrier Caps and Floors under the LIBOR Market Model with Double Exponential Jumps

The LIBOR market model (LMM) has become the standard model for many kinds of interest rate derivatives, such as cap contracts. It assumes that the distribution of the one-period interest rate at each future repricing date is lognormal, so that every caplet can be easily priced using the Black model. Volatilities at different dates are tied together by assumptions that restrict the number of parameters to a manageable set. But interest rates are strongly affected by things like monetary policy decisions, which can lead to sharp, nondiffusive jumps when policy changes. The effect may be seen in the leptokurtosis of the empirical distributions of interest rates and in volatility smiles for caplets. The low probability of extreme rate moves in the standard LMM is especially problematical for barrier options. In this article, the authors introduce a new jump-diffusion process for the LLM that can capture the effects of jumps on interest rate barrier contracts. The authors allow exponential jumps, both up and down, so the model is called the LMM with a double-jump process, or LMMDJ. Explicit valuation equations are provided for the four classes of barrier contracts and are shown to be quite accurate in Monte Carlo simulations.