sorry, **if you can create a model that will give you a better estimate than the current implied vol. Then you will get a better estimate of the instantaneous vol. You can still use BS, just change the input of the implied vol. Have you looked into Local vol? It is based on BS but gives you instantaneous vols. Which I think is what you are looking for? I am going to upload some plots here. Give me a few
Regressing SPX realized vol on to VIX gives us a an R^2 of .73 which is slightly higher than regressing the SPX change vs VIX change of .67. This plot will give you an idea of where the VIX will be given a realized vol. So what I am thinking is if SPX drops 50 points by tomorrow than the 30 day realized vol will = .40(an arbitrary #) and so therefor the VIX should = ~ 35 according to this model. PS this is data from 2010 to present. Vol is Yang Zhang not close-close. If I can remember you use R as-well. Data is available to 1993 with getSymbols if you are looking to do something more rigerous
While it's true that regressing VIX against 30D realized vol has much more explanatory power (I got an R^2 of 0.9 using close-to-close and data since 2004), the problem is this measure is slow moving and does not capture the up/downside asymmetry in SPX price changes. Seems like it would be useful for inference but not so much for forward looking estimates.
I've looked into the local vol model, and it's one way to do the "comprehensive approach" I described in my OP. You would use the local vol model to estimate changes in implied variance at each strike as a function of the SPX price change, and then use those numbers to re-compute the VIX. That would certainly be the ideal way to approach this, but as I mentioned before, it requires a lot more data, which is something I do not have. What I'm looking for is something that loosely captures the relationship between SPX and VIX without requiring all those parameters. In other words, a "sufficient statistic", if one exists. For now, it seems I will have to make do with the linear model.
This is really high, if you can find a way to incorporate downside/upside vol, you'll have a pretty good model. Let me know of your findings
Is this for academic purposes? If not, then may i ask what exactly are you trying to hedge? Obviously it would depend on purpose but I find simple linear modeling sufficient. Take the M1 or M2 future or some sort of mix and adjust for time to spot. For most practical purposes anything beyond that is an overkill, IMO.
I'm trying to determine a reasonable amount of SPX short deltas to hedge a short vol position. To clarify, do you mean regress VIX against M1 and M2 futures? How would this be used to determine the level of VIX corresponding to a hypothetical SPX price change?
In this case you won't be able to solve for the convexity without running the risk of overhedging. You just have to kind of pick your battles.
Could you explain why this would be overhedging? For instance, suppose I have a 50 delta VXX short. I want to hedge this with an SPX short position. Using the last 100 days of data, VXX has a beta of around -4 vs SPX. So, in order to hedge, I'd open a short position in SPX with -200 deltas. This already sounds like overhedging, but is this what you're referring to? If so, what's a better way to approach this in your opinion?
Because on the downside the beta is 6 and on the upside the beta is 3.5 and there are many more up days than down days. So either you are under hedged for a down day or overhedged in an up day.