Implied volatility as a quantile

Black–Scholes implied variance is the quantile of a generalized inverse Gaussian law, implied volatility its square-root transform. The insight follows from the known equivalence of normalized option prices and probabilities on the variance scale. The representation gives implied volatility a first-passage-time interpretation, ties the inverse Gaussian to the pricing heat equation, and turns the Greeks into derivatives of its density. The extreme-strike and near-forward asymptotics of the smile, previously derived by separate methods, emerge as the two tails of the same law, and its mode determines the curvature of the price-to-variance map.