Can There Be an Explicit Formula for Implied Volatility?
6 Pages Posted: 24 Nov 2012
Date Written: November 21, 2012
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.
Keywords: Call option, Black-Scholes formula, implied volatility, D-finite function, asymptotics
JEL Classification: G13
Suggested Citation: Suggested Citation