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In Gatheral's book, in the derivation of local volatility in terms of implied volatility, we use the regular Dupire formula

$$ \frac{\partial C}{\partial T} = \frac{1}{2} \sigma^{2}K^{2}\frac{\partial^{2}C}{\partial^{2}K} + (r_{T}-q_{T}) \left ( C - K \frac{\partial C}{\partial K} \right ) - r_{T}C $$

and reparametrize with $y = \ln \frac{K}{F_{T}}$ and $w = \Sigma(K,T)^{2}T$ where $\Sigma(K,T)$ is the implied volatility and get:

$$ \frac{\partial C}{\partial T} = \frac{v_{L}}{2} \left [ \frac{\partial^{2}C}{\partial^{2}y} - \frac{\partial C}{\partial y} \right ] + \mu_{T}C $$

where $v_{L}$ is the local variance.

Next, we set the market price equal to the B-S price to write the formula above in terms of derivatives of the B-S function:

$$ C(S_{0}, K, T) = C_{BS}(S_{0}, K, \Sigma(K,T), T) $$

So far so good.

But in the next step when we write $\frac{\partial C}{\partial T}$ in terms of the B-S call function, we get:

$$ \frac{\partial C}{\partial T} = \frac{\partial C_{BS}}{\partial T} + \frac{\partial C_{BS}}{\partial w}\frac{\partial w}{\partial T} $$

However, I would expect it to be

$$ \frac{\partial C}{\partial T} = \frac{\partial C_{BS}}{\partial T} + \frac{\partial C_{BS}}{\partial w}\frac{\partial w}{\partial T} + \frac{\partial C_{BS}}{\partial y}\frac{\partial y}{\partial T} $$

since a shift in maturity implies a shift in the forward and thus in the log-moneyness $y$. Gatheral's explanation is that it "follows from the fact that the only explicit dependence of the option price on T [...] is through the forward price". Why do we not also take into account the log-moneyness shift here?

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