THE MOMENT FORMULA FOR IMPLIED VOLATILITY AT EXTREME STRIKES

Source abstract

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.