An Explicit Solution to Black-Scholes Implied Volatility

TL;DR

This paper introduces an explicit, exact formula for Black-Scholes implied volatility using quantile functions and the inverse Gaussian distribution, improving computational speed and interpretability.

Contribution

It provides a novel, non-approximate explicit formula for implied volatility, rewriting the price-to-volatility map as a distributional transform based on variance quantiles.

Findings

Achieves machine precision faster than existing solvers.

Provides a new interpretative framework for implied volatility and Greeks.

Reorganizes no-arbitrage conditions in variance-quantile coordinates.

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.