I'm thinking ABM with heterogeneous agents... hoarders, traders etc.; you might run a toy model in StarLogo or any other such easy prototyping tool and if it looks promising you could try a bigger one with NetLogo and eventually go all out. But yeah, ABM or nothing, IMHO. This is not something that can be predicted by GARCH or whatever-traditional, in the short run.

Or, on account of the huge number of data points, you could run a two-variable Nadaraya-Watson with something likely (market sentiment as measured by some aggregate indicator?) and something unlikely (price of gold?) and see what happens.

I'm not even sure this suggestion makes any sense, it still looks too idiosyncratic to work, i.e. your ABM wouldn't have sensible behavioral rules no matter how hard you tried. And now of course you whetted my appetite for excessively nerdy stuff.