Holy Volatility Clustering!

So I'm sure some of you have read the comment about vol clustering by an astute reader of my kurtosis post. I figured I'd explain the idea of vol clustering a bit because, well, volatility continued just as the reader predicted.

Volatility clustering, just like it sounds, refers to the "clustering" of volatility. That is, large moves often come in packs. So when we saw our 2 standard deviation day, we expected more days with relatively large moves (and we continued to have 2 standard deviation intra-day moves every day since for four or five days now). Given volatility clustering, however, we would say that we have entered a new regime of higher vol, so these shifts aren't "that" big in stdev terms.

Volatility clustering is modeled by the ARCH and GARCH models. GARCH being the more advance and more used/known of the two. GARCH stands for generalized auto-regressive with conditional heteroskedasticity. A mouthful, eh? This is a type of time-series model. For those of you who would like to study time-series modeling/econometrics, the standard text is the Hamilton - "Time Series Analysis." Excellent book. You should have pretty strong math to go through it though.

Auto-regressive models are models that look back on previous periods' error terms (known as AR(n) models). The regression equation looks like [alright, I had to get rid of the equation b/c it was fucking up my formatting. E-mail me if you really want it, or look up an AR(2) model].

Heteroskedasticity is the state of having different volatility (standard deviation) describing the regression error. One of the primary assumptions of linear regression is homoskedasticity (the state of having constant standard deviation describing the regression error). Without homoskedasticity, linear regression is no longer efficient.

A regression is efficient if given any other estimate of the regression parameter, the efficient regression parameter converges to the real parameter faster. Actually, that's not really the definition, but it's close enough for those of you who don't know what it is. You should actually look up the proper definition, but it's a bit technical for me to write in a text blog (unless I post a LATEX document).

Anyway, now that we've defined the terms, what the hell does the GARCH model do? It allows us to say, "given that I just saw a shit-load of volatility, where do I expect my volatility to be going forward?" This is a rather helpful trait for modeling expected movements and even pricing options.

A lot of empirical papers have shown the existence of the volatility clustering phenomenon, and these last few days have also exhibited this behavior. It's interesting, to say the least. Might be worth someone's time to look at the effects of vol clustering and swaption implied vol. I've been meaning to because the swaption implied vol seems to be lagging the high vol effects. Then again, implied vol has historically been way to high in options pricing to begin with.