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Erschienen in:
2015 | OriginalPaper | Buchkapitel
Unifying the BGM and SABR Models: A Short Ride in Hyperbolic Geometry
verfasst von : Pierre Henry-Labordère
Erschienen in: Large Deviations and Asymptotic Methods in Finance
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Abstract
In this paper, using a geometric method introduced in (Henry-Labordère Large Deviations and Asymptotic Methods in Finance (2015) [12]) and initiated by (Avellaneda et al. Risk Mag. (2002) [4]), 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 (Hagan et al. Willmott Mag. 88–108 (2002) [11]). The geometry underlying this model is the hyperbolic manifold \({\mathbb H}^{n+1}\) with n the number of Libor forward rates.