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Suppose I have a volatility smile for a certain underlying at a given maturity. This implies a certain density for the underlying at that maturity, which can be explicitly computed, via a corollary to Dupire's formula.

Now suppose that I bump up the - say - right wing of the smile, i.e. all implied vols stay the same from $ K= 0 $ to $ K= F_t$, the forward, while the rest of the smile is bumped up a little.

What happens to the implied density of the underlying?

My reasoning was: implied vol goes up => call prices go up => higher probability that underlying price is higher => fatter right tail.

But from the put-call parity, also put prices go up => fatter left tail.

So, I would conclude that both tails increase, but this is at odds with the corollary to Dupire's formula, which is local, i.e. a bump in the right wing only changes derivatives there, so it would leave the left tail untouched.

What is wrong with my reasoning then?

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    $\begingroup$ Presumably the prices of Puts with high K do not have much of an effect on the left tail (which has more to do with the prices of Puts with low K). $\endgroup$ – Alex C Aug 29 '18 at 17:17

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