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

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.
Document type :
Preprints, Working Papers, ...
Domain :

Cited literature [19 references]

https://hal.archives-ouvertes.fr/hal-00015510
Contributor : Pierre Henry-Labordere <>
Submitted on : Sunday, January 22, 2006 - 11:59:19 PM
Last modification on : Monday, January 23, 2006 - 11:44:26 AM
Long-term archiving on: : Monday, September 20, 2010 - 2:06:03 PM

### Citation

Pierre Henry-Labordere. Unifying the BGM and SABR Models: A short Ride in Hyperbolic Geometry. 2006. ⟨hal-00015510v2⟩

Record views