VIX is to SPX as VVIX is to ?

The answers seems obvious - VIX is the 30-day volatility of SPX, VVIX is the 30-day volatility of VIX, but in reality it is a little more complicated than that, as least when it comes to the difference between historical and implied volatility, or volatility risk premium. After discussing this with one of the readers I decided to make this a short post.

VIX measures 30-day implied volatility of SPX options, and is a (granted, biased) predictor of future 30-day historical volatility of SPX index. However this same is not true for VVIX, that is VVIX is not a predictor of future 30-day historical volatility of VIX. It is however a predictor of future 30-day historical volatility of 30-day VIX forward. Such instrument does not exist, but the next best thing is VXX - ETF that holds front and second month futures to approximate returns on 30-day forward.

Something to note that theoretically term structure of implied volatility of VIX is decreasing with time to expiration, but VXX has a flat term structure, corresponding (approximately) to 30-day VIX implied volatility.