Abstract
Let denote the implied volatility at maturity t for a strike
, where
and
is the current value of the underlying. We show that
has a uniform (in x) limit as maturity t tends to infinity, given by the formula
, for x in some compact neighbourhood of zero in the class of affine stochastic volatility models. Function
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).
Notes
3. The name stems from the fact that under the numeraire asset is the risky security
.