One book said hedging variance swaps
$$I= \sqrt{\dfrac{1}{t}\int^t_0\sigma^2(S,t)}d t$$
by vanilla option
,say value $V(S,E;\sigma)$(Black-Scholes
fomula) where $S$ is underlying asset
, $E$ is the strike price
. Then he constructed a portfolio
with value
$$P=\int^{\infty}_0f(E)V(S,E;\sigma)d E$$
then compute the vega
of portfolio: $\textrm{Vega}_P= \dfrac{\partial P}{\partial\sigma},$
then let
$$\dfrac{\partial \textrm{Vega}_P }{\partial S}=0$$
obtain $f(E) = \dfrac{k}{E^2}.$
His conclusion is variance swaps can be hedged with vanilla option, using the 'one over strike squared' rule.
I can not understand:
1.What's the meaning of the representation of $P$(why take the integral) i.e how do we implement this portfolio by vanilla option, hold $F(E)$ share?
2.Why we need constant vega against $S$ i.e how to hedge?