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I'm trying to follow Gatheral 2006 on his derivation of the BSIV from a characteristic function. The most relevant formula is (5.7) page 60.

$$\int_0^\infty\frac{du}{u^2+(1/4)}\Re[e^{-iuk}\left(\phi_T(u-1/2)-e^{-1/2(u^2+1/4)\sigma_{BS}^2T}\right)]=0$$

He starts with a Proof of Lewis 2000 (5.6) by inversion. In the second part of his proof, he integrates a covered call position along the $Im[u] = 0.5$-line. Why does the covered call value only exist in the region $0<Im[u]<1$? Why did he choose to integrate along 0.5?

Furthermore, he substitutes the BS characteristic function and thereby finds a relationship between the ATM BS IV and the underlying process.

a.) Since the expression is dependent on log moneyness, why would he consider it only ATM? It would be my understanding, that this relationship should hold for the entire vol smile (given the same drawbacks of any other BS Vol Smile, and just being a more efficient way of quoting option prices across k)

b.) How would I numerically implement a.)? My approach so far is to minimize the left hand side of (5.7), but that includes multiple evaluations of the same integral, since my variant is sigma. Another way of thinking about it could be that, given a characteristic function, one would calculate option prices along k, and individually compute their BS IV (e.g. using Brent-Dekker), to receive a simulated vol smile.

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