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Paper detail

Can there be an explicit formula for implied volatility?

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.

preprint2012arXivOpen access
Stefan Gerhold
q-fin.PRmath.CA
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Stefan Gerhold

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