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No. Implied volatility isn't a historical measure of standard deviation. Implied volatility is used to relate a market price to some model, be that Black-Scholes or something more sophisticated. Another way to phrase it, implied vol is that single vol input into a model, such that the model reproduces the market prices. Different models will have ...


Since American style options allow early exercise, put-call parity will not hold for American options (unless they are held to expiration). In practice, there is also a difference between calls and puts for European options as well. The full description is here: What causes the call and put volatility surface to differ?


A pithy way to put it is "implied volatility is the wrong number to put in the wrong formula to get the right price." That is, implied volatility is by definition the parameter $\sigma$ to plug into the Black-Scholes option pricing formula to get the market price of a vanilla option. This is called "volatility," but in reality it isn't the same as the ...

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