Vanna–Volga Consistency Across Different Strike Sets

Summary

The document raises a theoretical question about a consistency property in the Vanna–Volga method. Starting from a price curve constructed by interpolation and extrapolation across strikes, the cited paper claims that selecting three different strikes on that same curve and rebuilding the curve with the Vanna–Volga procedure recovers the original curve exactly. The post asks whether this is an analytical theorem or only an empirical observation, since it does not find a proof in the paper.

No answer or derivation is included, so the document does not establish the result or identify a separate proof. Its value is in isolating a consistency question about how the method's reconstructed curve depends on the chosen reference strikes. The claim applies to the price curve and construction specified by the cited paper; the post does not discuss assumptions, boundary conditions, or whether the property extends to other interpolation choices or market setups.

Key ideas

• The post identifies a claimed invariance of a Vanna–Volga reconstructed price curve to the choice of three reference strikes.
• The cited paper states that rebuilding from other strikes on the original curve reproduces that curve.
• The document asks for an analytical proof but provides no proof or answer.
• The claim should be understood in the context of the price construction and assumptions described in the referenced paper.

Tags

# Vanna-Volga consistency result


# Vanna-Volga consistency result












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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