Can there be an explicit formula for implied volatility?
Pricing of Securities
2012-11-22 v1 Classical Analysis and ODEs
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.
Cite
@article{arxiv.1211.4978,
title = {Can there be an explicit formula for implied volatility?},
author = {Stefan Gerhold},
journal= {arXiv preprint arXiv:1211.4978},
year = {2012}
}