Can there be an explicit formula for implied volatility?

TL;DR

This paper proves that implied volatility cannot be expressed as a D-finite function of underlying parameters, indicating fundamental limitations on explicit formulas for implied volatility.

Contribution

It establishes that implied volatility is not a D-finite function, highlighting a key mathematical limitation on deriving explicit formulas.

Findings

Implied volatility is not D-finite as a function of underlying, strike, and call price.

The result constrains the class of functions that can represent implied volatility.

No explicit formula exists within the class of D-finite functions for implied volatility.

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.