Solving SABR in Exact Form and Unifying it with LIBOR Market Model

SABR stochastic volatility model is appealing for modeling smile and skew of option prices. Hagan, who first proposed this model, derived a closed form approximation for european options and showed that it provides consistent and stable hedges. Here I prove a new exact closed formula for the joint probability density of underlying and volatility processes, when correlation is zero. I argue that this formula remains a very good approximation when correlation is different from zero. I deduce from this expression different formulae for European options. After reviewing the Libor Market Model and its stochastic volatility extensions, I will show how to specify a unified SABR-LMM with a smile, where the term structure of skew is captured, and where closed formulae for caplets and robust approximations for swaptions are available.