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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 theunderlying. 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 somecompact neighbourhood of zero in the class of affine stochastic volatilitymodels. The function $h^*$ is the convex dual of the limiting cumulantgenerating function $h$ of the scaled log-spot process. We express $h$ in termsof the functional characteristics of the underlying model. The proof of thelimiting formula rests on the large deviation behaviour of the scaled log-spotprocess as time tends to infinity. We apply our results to obtain the limitingsmile for several classes of stochastic volatility models with jumps used inapplications (e.g. Heston with state-independent jumps, Bates withstate-dependent jumps and Barndorff-Nielsen-Shephard model).

Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models