  Let $\sigma_t(x)$ denote the implied volatility at maturity $t$ for a strike
$K=S_0 e^{xt}$, where $x\in\bbR$ and $S_0$ is the current value of the
underlying. We show that $\sigma_t(x)$ has a uniform (in $x$) limit as maturity
$t$ tends to infinity, given by the formula
$\sigma_\infty(x)=\sqrt{2}(h^*(x)^{1/2}+(h^*(x)-x)^{1/2})$, for $x$ in some
compact neighbourhood of zero in the class of affine stochastic volatility
models. The function $h^*$ is the convex dual of the limiting cumulant
generating function $h$ of the scaled log-spot process. We express $h$ in terms
of the functional characteristics of the underlying model. The proof of the
limiting formula rests on the large deviation behaviour of the scaled log-spot
process as time tends to infinity. We apply our results to obtain the limiting
smile for several classes of stochastic volatility models with jumps used in
applications (e.g. Heston with state-independent jumps, Bates with
state-dependent jumps and Barndorff-Nielsen-Shephard model).
