Can there be an explicit formula for implied volatility?

Stefan Gerhold

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arXiv · 1211.4978

Can there be an explicit formula for implied volatility?

Stefan Gerhold

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Abstract

It is "well known" that there is no explicit expression for the Black-Scholes implied volatility. We prove that, as a function of underlying, strike, and call price, implied volatility does not belong to the class of D-finite functions. This does not rule out all explicit expressions, but shows that implied volatility does not belong to a certain large class, which contains many elementary functions and classical special functions.

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Stefan Gerhold. 2012-11-21. Can there be an explicit formula for implied volatility?. https://arxiv.org/abs/1211.4978

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