An Explicit Solution to Black-Scholes Implied Volatility
This paper identifies what appears to be the first explicit formula for Black-Scholes implied volatility, resolving a 50-year-old problem in option pricing. The key observation is that the call price can be written as a survival probability of an inverse Gaussian distribution. Inverting this identity expresses implied volatility directly through the corresponding quantile function. The formula uses only observable option inputs and requires no initial guess, iterative inversion, approximation, asymptotic expansion, or infinite series. Numerical tests recover implied volatility to machine precision and show the formula to be about 3.4 times faster than a current state-of-the-art reference benchmark.
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Quantitative Finance > Mathematical Finance arXiv:2604.24480 (q-fin) [Submitted on 27 Apr 2026] Title:An Explicit Solution to Black-Scholes Implied Volatility Authors:Wolfgang Schadner View a PDF of the paper titled An Explicit Solution to Black-Scholes Implied Volatility, by Wolfgang Schadner View PDF HTML (experimental) Abstract:This paper identifies what appears to be the first explicit formula for Black-Scholes implied volatility, resolving a 50-year-old problem in option pricing. The key observation is that the call price can be written as a survival probability of an inverse Gaussian distribution. Inverting this identity expresses implied volatility directly through the corresponding quantile function.
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