In the GS Research Note about Volatility Swaps, it is shown that you can replicate a pure variance exposure (hedge) with only vanilla calls and puts, primarily thanks to the Carr-Madan formula of payoff reconstruction.
Derman et al. mention how using the log contract replication is a higher order term replication, so there are times where first and second order terms dominate (and vice versa). This is where the error comes into play.
This is the same formula used to calculate the VIX index - just in discrete form instead of continuous. What are the necessary implications of the discrete VIX calculation and this error term, and is there any method that is robust to jumps?
On page 30 of the report, it is mentioned that if the jump is "small enough" it can be considered part of the diffusion and has no impact on the variance. Can "small enough" be quantified a little further?