An Explicit Solution to Black-Scholes Implied Volatility

Wolfgang Schadner

Papers from arXiv.org

Abstract: This paper observes that the Black--Scholes call price can be written as the survival probability of an inverse Gaussian distribution, equivalently as a probability in variance space. Inverting this representation yields an analytically explicit formula for implied volatility in terms of the corresponding inverse Gaussian quantile function, with volatility on the left-hand side and only observable option inputs on the right-hand side. Numerical tests recover implied volatility to machine precision and, in a controlled setting, show the formula to be faster than a state-of-the-art benchmark.

Date: 2026-04, Revised 2026-05
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