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.

The tail of risk neutral returns can be related explicitly with the wing behaviour of the Black-Scholes implied volatility smile. In situations where precise tail asymptotics are unknown but a moment generating function is available we establish, under easy-to-check Tauberian conditions, tail asymptotics on logarithmic scales. Such asymptotics are enough to make the tail-wing formula (see Benaim and Friz (2008)) work and so we obtain, under generic conditions, a limiting slope when plotting the square of the implied volatility against the log strike, improving a lim sup statement obtained earlier by Lee (2004). We apply these results to time-changed exponential Lévy models and examine several popular models in more detail, both analytically and numerically.

In a recent article the authors obtained a formula which relates explicitly the tail of risk neutral returns with the wing behavior of the Black Scholes implied volatility smile. In situations where precise tail asymptotics are unknown but a moment generating function is available we first establish, under easy-to-check conditions, tail asymptoics on logarithmic scale as soft applications of standard Tauberian theorems. Such asymptotics are enough to make the tail-wing formula work and we so obtain a version of Lee's moment formula with the novel guarantee that there is indeed a limiting slope when plotting implied variance against log-strike. We apply these results to time-changed Lévy models and the Heston model. In particular, the term-structure of the wings can be analytically understood.

We survey recent results on the behavior of the Black-Scholes implied volatility at extreme strikes. There are simple and universal formulae that give quantitative links between tail behavior and moment explosions of the underlying on one hand, and growth of the famous volatility smile on the other hand. Some original results are included as well.

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