Can there be an explicit formula for implied volatility?

2012/11/21 by Gerhold, Stefan
#33E20 #91G20 #Classical Analysis and ODEs (math.CA) #FOS: Economics and business #FOS: Mathematics #Pricing of Securities (q-fin.PR)

paper · doi:10.48550/arxiv.1211.4978

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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