Unifying the BGM and SABR Models: A short Ride in Hyperbolic Geometry

Pierre Henry-Labordere
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Pierre Henry-Labordere: SOCIETE Generale

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

Date: 2006-02
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