Fabio Mercurio, Damiano Brigo, Francesco Rapisarda
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, existence and uniqueness results for the related SDEs and 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 option prices at any time. As an example, we analyze the linear combination of geometric Brownian motions with perfectly correlated (decorrelated) returns.
Examples of the implied-volatility curves produced by the considered models are finally shown.