An Explicit Solution to Black-Scholes Implied Volatility

Abstract

Black-Scholes implied volatility is a quantile. The insight follows from the normalized option price being a probability on the variance scale, with the inverse Gaussian distribution providing the link. It enables analytically exact and explicit formulas for implied volatility in terms of existing quantile functions, with volatility on the left-hand side and only observable option inputs on the right-hand side. The result is not another approximation or asymptotic expansion. Instead, it rewrites the price-to-volatility map itself as a distributional transform. The representation gives implied volatility a first-passage-time interpretation, identifies variance as the natural coordinate of inversion, and reorganizes Greeks and no-arbitrage restrictions in the same variance-quantile coordinates. Numerically, the formula achieves machine precision faster than a state-of-the-art solver in the benchmark considered. The paper therefore provides a new coordinate system for computing, interpreting, and decomposing one of the central quantities in option markets.