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An Explicit Solution to Black-Scholes Implied Volatility

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  • Wolfgang Schadner

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.

Suggested Citation

  • Wolfgang Schadner, 2026. "An Explicit Solution to Black-Scholes Implied Volatility," Papers 2604.24480, arXiv.org, revised May 2026.
    • Handle: RePEc:arx:papers:2604.24480
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    File URL: http://arxiv.org/pdf/2604.24480
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