The static replication result, i.e. a continuous strip of adequately weighted OTM vanillas can be used to replicate a variance swap, only holds:

• under a pure diffusion assumption
• one considers continously sampled realised variance which is the same as the annualised quadratic variation of the logarithm of the price process.

In practice these assumptions are not met due to

• presence of jumps in the underlying price process (error 1)
• discretely sampled variance in the specification of variance swap contracts (error 2)

Anyway, because a continuous strip of options is not practical from a pure trading perspective, it is replaced that by a finite sum (error 3). This is exactly what the VIX formula does.

It seems from your question like you are interested in both errors 1 and 3.

The paper "The Effect of Jumps and Discrete Sampling on Volatility and Variance Swaps" by Broadie and Jain discuss these, along with error 2, in the context of some particular stochastic volatility with jump model, see section 6.1 Effect of jumps on fair variance strikes. The references cited may also be useful. An electronic copy is available here for download.

The static replication result, i.e. a continuous strip of adequately weighted OTM vanillas can be used to replicate a variance swap, only holds:

• under a pure diffusion assumption
• if one considers continously sampled realised variance which is the same as the annualised quadratic variation of the logarithm of the price process.

In practice these assumptions are not met due to

• presence of jumps in the underlying price process (error 1)
• discretely sampled realised variance in the specification of variance swap contracts (error 2)

Anyway, because a continuous strip of options is not practical from a pure trading perspective, it is replaced that by a finite sum (error 3). This is exactly what the VIX formula does.

It seems from your question like you are interested in both errors 1 and 3.

The paper "The Effect of Jumps and Discrete Sampling on Volatility and Variance Swaps" by Broadie and Jain discuss these, along with error 2, in the context of some particular stochastic volatility with jump model, see section 6.1 Effect of jumps on fair variance strikes. The references cited may also be useful. An electronic copy is available here for download.