Pricing the smile in a forward LIBOR market model

We introduce two general classes of analytically-tractable diffusions for modeling forward LIBOR rates under their canonical measure. The first class is based on the assumption of forward-rate densities given by the mixture of known basic densities. We consider two fundamental examples: i) a mixture of lognormal densities, and ii) a mixture of densities associated to “hyperbolic-sine” processes. We derive explicit dynamics, prove existence and uniqueness results for the solution to the related SDEs and obtain closed-form formulas for caps prices. The second class is based on assuming a smooth functional dependence, at expiry, between a forward rate and an associated Brownian motion. This class is highly tractable: it implies explicit dynamics, known marginal and transition densities and explicit caplet prices at any time. As an example, we analyze the dynamics given by a linear combination of geometric Brownian motions (GBM) with perfectly correlated (decorrelated) returns. We finally construct a specific model in the second class that reproduces exactly the market caplet volatilities given in input. Examples of the implied-volatility curves produced by the considered models are also shown.