Variance Swap Revisited, Part I

1. Why is it that Vega Notional = 2K *Variance Notional? Because when calculating P&L, (sigma^2 - K^2)/2K is roughly "d sigma". This approximation, of course, is linear and is good only when close to strike, which brings us to...
2. CONVEXITY. Variance swap payoff is convex with respect to volatility. The further away realized variance is from the strike, the more pronounced the effect of convexity is. This means that 1) the fair strike for a variance swap is always greater than the fair strike of a comparable volatility swap because of the benefit of convexity; and 2) Variance swap price increases with vol-of-vol, but volatility swap price does not.
3. The forward arithmetic of IR is mutiplicative; the forward arithmetic of variance is additive.
4. For capped variance swaps, unwinding by entering into an opposite trade is complicated by the fact that the cap itself would move when one tries to lock in the P&L after some time, causing the offsetting trade to have a different cap level than the original trade.
5. Derman's approximation assumes that implied volatility curve is linear. This approximation is good when vol-of-vol effect is low, e.g. for shorter-term index variance swap (instead of single stock variance swap)
6. Volatility risk premium: the price of variance swap is 'too high' as compared to both historical realized volatility or implied vanilla option volatility. In other words longing variance is in high demand. Not to be confused with the vol-of-vol effect.
7. Variance swap can be replicated by OTM call and put options, hence the fair strike is a weighted average of implied variances across the entire volatility smile, which, given a volatility smile (going up both ways), is greater than sigma_ATM.

Reference: JP Morgan Variance Swaps