What theoretical distribution best fits the vix?
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$\begingroup$ Real world? Risk neutral? 'Best' is a loose or ambiguous concept. You could fit the VIX distribution using a mixture of distributions (eg mixture of lognormals), shifted lognormal perhaps, or given a number of moments of the VIX you could fit some other distribution such as Pearson. I don't think there is a unanimously agreed 'best' distribution. $\endgroup$– FridoSep 29 at 15:28
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$\begingroup$ To give more detail, I'm trying to normalise the difference between the VIX and the 9 day VIX. I'd like to do this with an algorithmic which doesn't need to be "fitted" (e.g. not something like quantile normalization). I'm wondering if there is some simple theoretical distribution which approximately fits the vix (in the same way that returns can be modelled by a normal distribution), so I can apply some analysis to work out the distribution of the difference, then see if there are some obvious ways to make it more normal. $\endgroup$– LmnopSep 29 at 16:16
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1$\begingroup$ A shifted lognormal distribution without drift can be a relatively good fit to options on the VIX. So for the VIX you could try a shifted lognormal with drift (since the VIX is not a martingale, in contrast to the VIX future). This is under risk-neutral though. I'm not sure what would be good under real-world. $\endgroup$– FridoSep 29 at 17:04
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