  In this paper, we derive a general asymptotic implied volatility at the
first-order for any stochastic volatility model using the heat kernel expansion
on a Riemann manifold endowed with an Abelian connection. This formula is
particularly useful for the calibration procedure. As an application, we obtain
an asymptotic smile for a SABR model with a mean-reversion term, called
lambda-SABR, corresponding in our geometric framework to the Poincar\'{e}
hyperbolic plane. When the lambda-SABR model degenerates into the SABR-model,
we show that our asymptotic implied volatility is a better approximation than
the classical Hagan-al expression . Furthermore, in order to show the strength
of this geometric framework, we give an exact solution of the SABR model with
beta=0 or 1. In a next paper, we will show how our method can be applied in
other contexts such as the derivation of an asymptotic implied volatility for a
Libor market model with a stochastic volatility.
