The Moment Formula for Implied Volatility at Extreme Strikes

Consider options on a nonnegative underlying random variable with arbitrary distribution. In the absence of arbitrage, we show that at any maturity T , the large-strike tail of the Black-Scholes implied volatility skew is bounded by the square root of 2|x|/T , where x is log-moneyness. The smallest coefficient that can replace the 2 depends only on the number of finite moments in the underlying distribution. We prove the moment formula, which expresses explicitly this model-independent relationship. We prove also the reciprocal moment formula for the small-strike tail, and we exhibit the symmetry between the formulas. The moment formula, which evaluates readily in many cases of practical interest, has applications to skew extrapolation and model calibration.