I have been thinking about this for a while... I can't make my head around it because of the gap that there's still on between financial economics and quantitative finance. Usually, when a student is introduced to the CAPM in undergrad for didactical purposes, she's told that the $\beta$ in the regression represents the sensitivity of an investment to an ideal "market portfolio". But assuming that the market portfolio is a weighted linear combination of all existing securities, it still does incorporate volatility clustering phenomena. This may lead to a guess that is not possible to eliminate volatility, and by this I mean even volatility clustering - i.e. persistent autocorrelation of the absolute/squared returns.
However, the anecdotical evidence from friends of mine working in hedge funds, is that it is actually possible to eliminate the volatility clustering phenomena with proprietary trading strategies. So my question is: if we consider the tail dependency in financial markets, we observe that in turbulent market times even uncorrelated assets start moving together, as noted by Bouchaud; this observation may lead to think that volatility is not a specific risk that can be eliminated through diversification. On the other hand, proprietary trading desks can set up properly their strategies so to avoid dependence on the second moment of their returns, and offer their investors more attractive returns. But where volatility stands in this framework? If it is possible to build a portfolio with no autocorrelation on the squared returns, I think that there's a good chance of dealing with alpha in its purest form, not some kind of "smart beta" professed in business school. This would lead me to think that volatility is a systemic risk for the ones that are not able to handle it, and something that can be eliminated for those who know what they're doing. Unfortunately, I haven't found any paper on the argument, but feel free to express your point of view, and even share papers on the subject, discussing a little bit what's in them.