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(I've gone through many questions/posts on quant StackExchange and only find responses about Black-Scholes)

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    $\begingroup$ Yes. The VIX (Chicago Board Options Exchange Volatility Index) is a measure of implied volatility calculated from a model-free formula (not relying on the Black Scholes model, or another model). This formula is somewhat advanced and may not be covered in introductory classes on options, which may explain why you have not seen it yet. Another disadvantage is that it requires the prices of options at all strikes not just one strike like BS IV, Why do you ask? $\endgroup$ – Alex C Sep 6 '18 at 17:30
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    $\begingroup$ When people say "implied volatility" they almost always mean Black Scholes Implied Volatility. To specify the other you have to say "model free implied volatility". There are posts on this site that discuss both. $\endgroup$ – Alex C Sep 6 '18 at 17:41
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    $\begingroup$ As Alex C wrote, with implied volatility it is silently understood that we mean the volatility parameter value to use with the Black-Scholes formula to recover the market price. With negative rates people now also make the distinction between normal implied vol (Bachelier) and lognormal implied vol (Black-Scholes) $\endgroup$ – ilovevolatility Sep 9 '18 at 9:47
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Yes, there is. Black Scholes is the formula for log-normal distribution.

Normal distribution has two parameters - the drift and the volatility. Under options theory (no arbitrage etc.), the drift is the risk free rate, and hence “known” per se. Thus, you pretty much end up with one equation / one variable to solve for.

But you can extent it to anything. In a previous firm, we derived IV using options formulas under

  1. log-hyperbolic secant distribution (2-parameter, of which in the option formula one will replace the “equivalent” drift by relating it to risk-free rate)

  2. log-Cauchy distribution(where the Cauchy distribution had a scale and location, i.e. 2-parameters )

  3. log-t distribution with fixed degree-of-freedom like 3, 5 etc (reduced 3-parameters to a 2-parameters by fixing the “shape” / df parameter)

  4. log generalized normal by fixing the shape of 3 parameters (and replacing the “equivalent” drift by relating it to risk-free rate)

E.t.c.

Not sure how many of these would be academically valid, but unlike the stochastic volatility or jump diffusion models that have many more parameters, all these models above have just 1 variable to solve for - and can be analytically inverted.

Some of these gave very interesting and dramatically different shapes versus the log-normal (Black Scholes) implied IV.

If you think outside the box of standard academic arguments , and consider it as a more “statistical arbitrage” problem, it is really one equation / one variable. Do whatever you wish to do in terms of the equation you want to solve. Some of the solutions are very tradeable (high Sharpe ratios ), when BS shows a high IV between two points, while some other models shows a low IV between two points etc.

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