For the purpose of getting fatter tails than the Guassian, I have seen people for example use $\alpha$-stable processes to model the stock. But in that case they end up using 'tempered' versions of the processes, where the tails decay exponentially so as to make the second moment finite. So the standard seems to be that the second moment must be finite. But why is this so? Is it just for tractability of the model or do they believe that finite second moment is an empirical fact?
In this paper Taleb explores the possibility of constructing a risk-neutral measure in an infinite-variance setting. As he mentions, this destroys the dynamic-hedging theory, but in practice it does not make a difference.