I know that a way of computing the price of a derivative paying $S^2$ at time $T$ is by making use of the following strategy:
$V=\int_{0}^{\infty} s^2 \frac{\partial^2 C}{\partial K^2}(K=s)ds$
Where $\frac{\partial^2 C}{\partial K^2}(K=s)$ is simply the risk neutral distribution of $S$.
Now, one should apply integration by parts twice in order to get an integral of the call prices. My question is, what is the name of this strategy/formula? Where can I find a derivation of it?
EDIT: By integrating by parts once, I get:
$V=\left[ s^2 \frac{\partial C}{\partial K}(K=s) \right]^{\infty}_{0} -\int_{0}^{\infty} 2s \frac{\partial C}{\partial K}(K=s)ds$
The first term is zero, but I don't know why. Can you please provide an explanation? Now by integrating by parts a second time the first derivative should become simply the price of options. But as I don't know why the first term is zero I hesitate to continue the derivation. Can you please help with next steps? EDIT: By taking into account the answer provided, the first term is zero,
$V= -\int_{0}^{\infty} 2s \frac{\partial C}{\partial K}(K=s)ds$
then we integrate by parts a second time:
$V= -\left(\left[2kC(k)\right]_{0}^{\infty}-\int_{0}^{\infty} 2 C(s)ds \right)$
Now, as
$C(k=\infty)=0$
Then:
$V= 2\int_{0}^{\infty}C(K=s)ds $
Is that correct?