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In the Vanna-Volga (VV) paper by Castagna and Mercurio they state that, once you build up a curve of prices by interpolating-extrapolating on $K$, you can recover the same exact curve by redefining the curve, now using three prices result of using three new different strikes and the VV method:

We now state two important consistency results that hold for the option price (7) and that give further support to the VV procedure. The first result is as follows. One may wonder what happens if we apply the VV curve construction method when starting from three other strikes whose associated prices coincide with those coming from formula (7). Clearly, for the procedure to be robust, we would want the two curves to exactly coincide. This is indeed the case. [...]

It's given as a result but its not proven within the paper. Is it just an "empirical" result? Does someone know if this result has been proven analytically somewhere?

Thanks!

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    $\begingroup$ The proof is given in Castagna and Mercurio, Consistent pricing of FX options. (Risk articles sometimes omit proofs given constraints on length etc). $\endgroup$
    – user34971
    Apr 15, 2022 at 13:59
  • $\begingroup$ Thanks Frido, I'll check it out! $\endgroup$
    – KT8
    Apr 15, 2022 at 16:50
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    $\begingroup$ Always happy to help out a fellow amateur quant (from your profile) :) $\endgroup$
    – user34971
    Apr 15, 2022 at 17:03

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