Hi everybody, is it possible to calculate a volatility surface from a volatility index like VIX?I know that Exchanges calculate a volatility index from listed option prices but I would like to solve the inverse problem: given a daily time series of volatility (the index) I would like to reconstruct the volatility surface that was used to calculate the index. Any paper or formula for doing that?Thanks all.

I don't think it is possible.

Why not? Please explain...Other ideas?

look at the VIX white paper.

One way you could go about it is to use a model for equity option prices, compute the value of the volatility index as a function(al) of the parameters of the model, calibrate the parameters to time series of volatility index (and probably equity index), and then compute volatility surface from the calibrated parameters. I am afraid that it won't work very well though because VIX probes only a small portion of volatility surface. You will probably have to include other instruments, like volatility futures. Check "Smile Dynamics" paper set by Bergomi, maybe you can get something out of it. Very roughly, looking at time series of VIX and SPX you can quickly learn volatility level (VIX), skewness (correlation between VIX and SPX) and kurtosis (volatility of VIX index), which should give you some idea about the vol surface of SPX, but in order to make it more precise you'll have to have a model.

VIX tells you the exactly and only the variance of the distribution. To get a vol-surface, you need more information: A vol surface tells you all the foward PDFs.

What about if I have the ATM term structure?Can I get from it the whole surface? Is there any model?Maybe you can fit some parameters to get the skew from an old surface and then use these parameters to modelize the whole surface starting form the ATM term structure...Any papers?

The VIX is like (but not equal to) a weighted average over all implied vols of a given expiration. Knowing an average doesn't tell you full info about what you have averaged over. You are still missing information about the skew at least. Knowing the variance and the ATM vols helps, as you can get an idea of the vol convexity, but the skew info is fully missing still. Some rough information about that might come from looking at the correlation between the SPX and the VIX, but it would be rough information. There are papers out there on how to get an approximate vol surface knowing the ATM, vol convexity and vol skew.Why do you need to do this?

I want to find a proxy to replicate a volatility surface in case I have no data at time T. I will use this proxy volatility surface for VAR calculation purposes.Case1): I have the whole surface at time T-1 but I have just the ATM term structure at time T--> how can I estimate skew from the surface at time T-1 and apply the estimated skew to the ATM term structure at time T to imply the whole missing surface at time T?Case 2): at time T I have no volatility surface and no ATM term structure --> I can calculate a simple different point by point in the volatility surface between T-1 and T-2 and sum this difference point by point at the volatility surface at time T-1 to get the surface at time T.What do you think? Any suggestion for case 1)? Is the method explained for case 2) good enough for VAR calculations?

If your portfolio has a material vega content then you need to do a better job than simple extrapolation of the vol surface. Your VaR will have a component which comes from the historical volatility of implied volatility. If you simply extrapolate the vol surface in a "non-volatile" way you will underestimate your VaR.There is probably not a simple answer here. If you have a full history of the vol surface you could do something like convert this information to "vol by delta" format keeping for example 10/25/50/75/90 delta implied vol time series. Once you see how these series behave (they will have some volatility and be correlated) you can potentially fill in missing data by simulation, but you would need to do that in a way that maintains the correlation structure between implied vol series, and also the index or stock itself.If the data looks horrible and noisy you could fit a model (through time) like the SVI (there is a "surface SVI" model as well for a whole surface) and this should keep the noise level down.That way you could create a defensible history of implied volatility for your VaR calculation.But just some simple extrapolation would underestimate VaR.

Thank you for your reply but I need to solve my problem as I explained it.I cannot go throught portfolio vega, I just need to have a proxy volatility surface in case I cannot calculate the true one and this proxy should come from what I told you before.

Let's discuss it in terms of SPX/VIX although the same considerations apply to other indices and their vixes.In principle, one could have dynamics (meaning a Q-process evolution model) such that the entire SPX implied volatility surface at time t was determined by the value of VIX(t,T), the standard vix for T=30 days. The CBOE also calculates a term structure of vixes for different T's, so a richer model would say that the VIX(t,T) termstructure determines the time-t implied volatility surface. You should test the two models above by historical fitting and then make a judgment as to whether or notthey are close enough to reality for your purposes. For example, by fitting Gatheral's SVI model to implied vol surfaces.This is similar to erstwhile's suggestion.My two cents.

Another approach might be a two step one: first, use Dupire's "breakeven vol" method (function BEVL in Bloomberg) to produce a vol surface for the SPX purely from SPX returns. Second, modify this structure in a way that forces it to fit the variance swap. Simple linear scaling is the simplest approach.If this ends up being too noisy, again fitting SVI would be a reasonable way of calming it down.