  We study here the large-time behaviour of all continuous affine stochastic
volatility models (in the sense of Keller-Ressel) and deduce a closed-form
formula for the large-maturity implied volatility smile. Based on refinements
of the Gartner-Ellis theorem on the real line, our proof reveals pathological
behaviours of the asymptotic smile. In particular, we show that the condition
assumed in Gatheral and Jacquier under which the Heston implied volatility
converges to the SVI parameterisation is necessary and sufficient.
