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. 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?
I'm not sure that there is a model. I've seen this happen -- the first time price drops to a level, VIX spikes high, then price bounces off the level and retests it the next day or a few days later and VIX won't jump to the same previous level. Regardless, even if you had an accurate model of VIX in relation to SPX moves, how would you use it? If you can accurately predict where the market is going, you don't need to do anything complicated to make money.
This is what I'm talking about when I say that the linear model cannot capture non-linearities. On Oct 11 the VIX hit a high of around 29. On Oct 23, SPX broke below the Oct 11 low but VIX peaked at around 25. I agree. The model I'm looking for would be useful for hedging.
Why would you then run a linear model instead of non-linear? You can just capture non-linear correlation, using any polynomial curve-fitting for example (using any depth you like). Often such models will be overfit though, especially if you try to get more exact fit. Or use copulas. But such models will substantially differ during any period you measure since, as ET180 pointed out, VIX often behaves independently of SPX. Sometimes it moves together with SPX, most of the time in opposite direction but sometimes it has delayed reaction, sometimes moves ahead. Those correlations were very different last February than they are in October, while they may change from day to day as well. Though when VIX is low then you may be able to project some movement up, without having to play a psychic in terms of SPX. We all know that when VIX is at 12 then at some point it will be at 20. We also know that it won’t last long above 50. You can deduce all this just by looking at VIX chart itself.
I agree this regression will not have any explanatory power. Another thing to keep in mind is vol has a time component, the index price does not. A 50 point move in SPX in 5 min will have a very different effect on the VIX than a 50 point move over 5 days. You want to compare apples with apples. If you want to calculate instantaneous moves local vol can help you out here. I would also think that regressing changes in realized vol against changes in VIX would help you out here as well ( although you would need to find a way to distinguish between upside vol and downside vol in your model as they will have a different effect on the VIX). Another tricky thing to keep in mind is that when VIX gets to crazy levels (both very high and very low) VIX and SPX can become correlated.
Adding a quadratic term to the model is insignificant (R^2 is virtually unchanged). I understand that, to some extent, VIX has a mind of its own and moves independently of SPX. When VIX is low it can stay low for an extended period of time (2017). When VIX is high it doesn't mean it has to mean revert immediately. I'm not trying to predict the VIX here. I'm just trying to look at what-if scenarios in a more rigorous way. A model that would answer questions like - If SPX drops 50 points, what would be a reasonable range for the VIX? A better model would give you a narrower range.
I think your answer is the local vol model. Given an implied volatility surface. If the index/stock moves to X right now, what will the implied volatility be. It is not too hard to back out the vix from the implied volatility.
sorry for hijacking this thread but I forgot to mention, if you have a better model than implied volatility then your local vol output will also be much better. GL
Can you elaborate on what you mean by "a better model than implied volatility"? I assume it means a model that's not Black Scholes.
The simple linear model actually has decent explanatory power, but regressing changes in VIX against changes in realized vol does not, although I haven't considered the asymmetry between down/upside vol. The local volatility model could be something to look at, thanks.