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Furthermore, assume that the current price of the underlying asset equals the strike price of the options.

If volatility measures variance without a direction, it doesn't make sense to me that the volatility would be different for a call than for a put with the otherwise same variables.

All this reasoning makes me conclude that the IV should be equal for the call and the put in this case.

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  • $\begingroup$ European or American. For the latter, it's perfectly reasonable that IV isn't identical, also theoretically. $\endgroup$
    – AKdemy
    Commented Apr 20 at 16:36

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In theory, they shouldn't, but in the real world, it is possible.

In Theory: Call and put options of the same strike and expiry should obey put call parity and thus have the same IV.

In Practice: Call and put prices should obey put call parity due to potential arbitrage opportunities but sometimes there are arbitrage costs such as bid-ask spreads that are unnecessarily wide that make the arbitrage unprofitable. Therefore, they can have (slightly) different IVs.

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