  We consider a class of asset pricing models, where the risk-neutral joint
process of log-price and its stochastic variance is an affine process in the
sense of Duffie, Filipovic and Schachermayer [2003]. First we obtain conditions
for the price process to be conservative and a martingale. Then we present some
results on the long-term behavior of the model, including an expression for the
invariant distribution of the stochastic variance process. We study moment
explosions of the price process, and provide explicit expressions for the time
at which a moment of given order becomes infinite. We discuss applications of
these results, in particular to the asymptotics of the implied volatility
smile, and conclude with some calculations for the Heston model, a model of
Bates and the Barndorff-Nielsen-Shephard model.
