2006/02/15 by Henry-Labordere, Pierre
#Computational Finance (q-fin.CP) #FOS: Economics and business #FOS: Physical sciences #Other Condensed Matter (cond-mat.other) #Physics and Society (physics.soc-ph)
paper · doi:10.48550/arxiv.physics/0602102
In this short note, using our geometric method introduced in a previous paper \citephl and initiated by \citeave, 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 \citesab. The geometry underlying this model is the hyperbolic manifold \HHn+1 with n the number of Libor forward rates.