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In this short note, using our geometric method introduced in a previous paper\cite{phl} and initiated by \cite{ave}, we derive an asymptotic swaptionimplied volatility at the first-order for a general stochastic volatility LiborMarket Model. This formula is useful to quickly calibrate a model to a fullswaption matrix. We apply this formula to a specific model where the forwardrates are assumed to follow a multi-dimensional CEV process correlated to aSABR process. For a caplet, this model degenerates to the classical SABR modeland our asymptotic swaption implied volatility reduces naturally to theHagan-al formula \cite{sab}. The geometry underlying this model is thehyperbolic manifold $\HH^{n+1}$ with $n$ the number of Libor forward rates.

Unifying the Bgm and Sabr Models: a Short Ride in Hyperbolic Geometry