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

Pierre Henry-Labordere (1)

1 SOCIETE GENERALE - Equity Derivatives Research Societe Generale

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.

Domaines: Physique et Société [physics.soc-ph]; Autre [cond-mat.other]

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