Volatility of Average

Different commodity including some volatility and bitcoin futures settle to an average value. For example VSTOXX settles to 1 minute sampled average value of the index in the last half hour of trading, Bitmex XBT/USD futures settle to 1 minute sampled average value of the last two hours of trading, Atlas ATS BTC/USD futures and options settle to a 24 hour average of the last calendar day of the month.
Correct modelling of this is interesting for both options and futures pricing, as average process has obviously less volatility than "regular" process. How much less? Well, apparently I fell asleep during a class because I have no recollection of the formula until I recently researched it.
B(t) is ABM, then has variance of


In simple terms, average has 1/sqrt(3), or about 58% of volatility. 

Simple example: assume BTC/USD volatility of 100% per year. Expected volatility over 30 days should be  sqrt(100 * 100 * 30/365)  = 28.67%. However if the contract settles to the last day's average, for options pricing we should use volatility of sqrt(100 * 100 * 29/365 + 100 * 100 * 1/365 / 3)  = 28.35%, slightly lower. 

In conclusion: average settlement - interesting, but of little practical importance from pricing perspective. 

P.S. Re: Atlas ATS - could not get their API to work, waiting for LedgeX.