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Hello people of the internet

I am doing a little experiment where I want to examine how well the VIX predicts the realized volatility, and how well the VVIX (Volatility of VIX Index) predicts the realized volatility of volatility. My questions are at the end of the post, I first try to explain what I did. In a first step, I calculated the 21 day real vol. of the SP500 with the equation below:

$$ RV_{t} = \sqrt{\frac{252 * \sum_{i=0}^{N-1}(ln(\frac{PX_{t-1}}{PX_{t-1-i}}))^{2}}{N}} * 100 $$

So in this case, $PX$ is the return of the SP500 Index. The two graphs visualize the relationship and my results seem to align with other results I have found online. The VIX level is usually higher than the realized volatility, which is due the excess of demand from market participants seeking the insurance like characteristics that options can provide. The two are highly correlated and the VIX therefore seems to be a good indicator of realized volatility. enter image description here Now the second, trickier part: Considering that the VVIX is calculated in an identical manner as the VIX (only difference being that VIX options are used instead of SP500 options), I thought that I could do the exact same experiment as above, but now with VVIX and realized volatility of volatility. So I calculated the realized vol of vol with the equation above, but with $PX$ now being the return of the VIX: Real LVoV stands for Realized  the cVolatility of Volatility So it seems as if the VVIX (and therefore the implied volatility information from Vix options) isn't as good at predicting the realized vol of vol.

So now (finally), my questions are:

  1. Is my approach and line of thinking correct, or did I oversee something? I am especially unsure if using equation 1 and VIX returns really gives me the actual real vol of vol.

  2. If my approach is correct, would it be true to say that the second results indicate that market participants are unable to predict volatility of volatility, or that the informational content on volatility of volatility in Vix options is more limited than information on volatility in spx options?

Sorry for the long post, I am an undergrad student and I am very thankful for any answer, recommendation, etc.

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