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Nevermind, i'm just confusing myself. Now I understand what I misunderstood.
The implied volatility surface of a prices of calls generated by a stochastic volatility model will not be constant since the implied volatility is found using the Black-Scholes model. The Black-Scholes model and a stochastic volatility model of course disagree on prices, and hence ...
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I assume you work in the Black Scholes framework. Then,
\begin{align*}
P(S_0,K,T) = Ke^{-rT}\Phi(-d_2)-S_0\Phi(-d_1),
\end{align*}
where
\begin{align*}
d_1 &= \frac{\ln\left(\frac{S_0}{K}\right)+\left(r+\frac{1}{2}\sigma^2\right)T}{\sigma\sqrt{T}}, \\
d_2 &= \frac{\ln\left(\frac{S_0}{K}\right)+\left(r-\frac{1}{2}\sigma^2\right)T}{\sigma\sqrt{T}}= ...
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The first part of your question:
$\frac{\partial y}{\partial S} = \frac{\partial ln S}{\partial S} = \frac{1}{S}$
$
\frac{\partial^2 V}{\partial S \partial y} =
\frac{\partial}{\partial y} \frac{\partial V}{\partial S} =
\frac{\partial}{\partial y} (\frac{\partial y}{\partial S}\frac{\partial V}{\partial y})=
\frac{\partial}{\partial y} (\frac{1}{ S}\frac{\...
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