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What is the reason that the vertical volatility skew graph(decreasing IV as the strikes increase) is the same for the puts and calls? The loose explanation is because of put call parity, but I am not able to find any more details.

I am looking for an explanation that makes sense intuitively.

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put call parity guarantees that the implied volatility of a call and put with the same strike is the same. So the smile graph is the same as well and so are all quantities derived for it.

In more detail, $$ C(K) = P(K) + F(K) $$ The value of $F(K)$ is model independent and does not depend on volatility. So knowing the implied of $C(K)$ gives you the price of $P(K)$. The same implied vol will work for $P(K)$ since put-call parity works for all models so it must hold for the BS model with that vol.

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