I have a doubt regarding the varswap replication- I know the portfolio of options with proper weights is a static one, and that there is a dynamic position required in underlying. My confusion is whether this dynamic position in underlying relates to delta hedging of the options i.e. are you required to calculate net delta of your portfolio of options and hedge that? I think it somehow relates to delta hedging as we are finally able to capture difference between implied and realized vol, but cant get my head around it.
Well I've never actually traded a var swap but I see no answers so I'll give it a shot. If you are short a var swap, your hedge is to buy options of all strikes. In practice, you buy a ATM straddle and some otm puts of various strikes and some otm calls of various strikes. Initially, this is close to delta neutral. Suppose the payoff of the variance swap is calculated from daily observations of the underlying. Then, you need to get back to delta neutral at the close of business on each day. Thus, if the market has gone up, your portfolio of hedge options has gotten long the market (since it is long gamma), so you need to sell the market to get back to flat. You do this every day. At the end of the trade your hedging activity has realized the actual variance of the stock during the period, which was your goal.
Take a look at the following note which explains that indeed the dynamic position in the underlying is from delta hedging the options.
I actually still have to complete the note with showing that the aggregate delta of the options portfolio is $1/S_t$, but I have not had time yet.