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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 ...

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