  We consider risk-neutral returns and show how their tail asymptotics
translate directly to asymptotics of the implied volatility smile, thereby
sharpening Roger Lee's celebrated moment formula. The theory of regular
variation provides the ideal mathematical framework to formulate and prove such
results. The practical value of our formulae comes from the vast literature on
tail asymptotics and our conditions are often seen to be true by simple
inspection of known results.
