I've been trying to gain a better understanding of variance swaps and what better way than to replicate it with a portfolio of better understood instruments.
I have read the GS 1999 paper and JPM 2005 paper and think I get how the replication works.
With the assumption of no jumps and full continuous strikes, the replication using options is exact, so the variance swap and the ideal replication portfolio should be indistinguishable. Now we know that the variance swap's variance Vega (d price/d variance) at any given time is simply the variance notional regardless of spot and vol level. So it follows that if I sum (do an integral across strikes) of the options' variance vegas, I should get a constant as well. However, in BS, Vega, gamma, variance Vega are all functions of vol. And vol is NOT a constant function of strike. Does this not mean that a different vol curve would generate a different Vega variance curve? Something is obviously wrong with my argument, but where is it wrong?