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

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).