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The SABR implied volatility is often used as an input in Black's model to price swaptions, caps, and other interest rate derivatives.

I'm wondering whether you can use the SABR closed form solution of the implied volatility curve as input into Black-Scholes European Call option formula?

Do you know any articles that uses SABR's implied volatility as input into the Black-Scholes equation to price european equity call options? Do you see any problems doing that?

Thank you in advance!

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    $\begingroup$ What is the reason for your doubt that plugging in the implied volatility that comes out of the SABR model into the Black-Scholes formula would lead to problems? Remember that implied volatility is by definition a number you need to plug into Black-Scholes (or Black) formula to obtain the market price. $\endgroup$
    – Frido Rolloos
    Oct 28 '20 at 13:34
  • $\begingroup$ @ilovevolatility since the derivation of the implied volatility in the SABR article is derived from Black' model - and not the Black-Scholes - it might results in some problems? $\endgroup$
    – Modvinden
    Oct 28 '20 at 15:48
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    $\begingroup$ Black and Black-Scholes is really the same model, just expressed a little differently. The volatility is the same. $\endgroup$
    – Jesper Tidblom
    Oct 28 '20 at 17:26
  • $\begingroup$ Also, as far as I know there are no known explicit formulas for the implied vol in the Sabr model. You have, the often used, Hagan approximative formulas. This is just an approximation which can become really bad for slightly more extreme input. $\endgroup$
    – Jesper Tidblom
    Oct 28 '20 at 17:29
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    $\begingroup$ I would recommend to use normal normal vol instead of lognormal vol when using SABR (we input the normal vol in the Bachelier model, not Black model) $\endgroup$
    – user39119
    Oct 28 '20 at 17:56

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