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I'm looking for a model that computes the expected VIX based on SPX price moves. For instance, SPX is currently at around 2650, and VIX is at 24. If SPX jumps to 2600, at what level would the VIX be?

I imagine that the comprehensive approach would be to model the changes in implied vols at each strike of SPX following the jump, and then recompute VIX based on the projected vols. That's great, but it requires a lot of data and is probably overkill for my purposes. At the simpler end of the spectrum, we can run a linear model of VIX changes against SPX changes, which would capture the negative correlation between the two indices, but would omit non-linearities. Are there any models that provide a more realistic approximation?

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