November 19th, 2014, 3:57 pm
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.
Last edited by EBal
on November 18th, 2014, 11:00 pm, edited 1 time in total.