August 25th, 2017, 2:38 pm

Seriously, while it's true that markets are highly efficient, your statistical statement is kind of muddled and likely easily rejected by simple tests. By "returns are randomly distributed", I will guess you mean returns [$]x_t[$] behave as if drawn independently each period (say day) from some fixed distribution -- so called i.i.d. draws. Then [$]S_T = \sum_{t=0}^T x_t[$] is a classic generic random walk, as studied by Frank Spitzer, for example.

But if that was true, not only would one find that [$]Corr(x_t,x_{t-1}) \approx 0[$] -- which one does -- but [$]Corr(|x_t|,|x_{t-1}|) \approx 0[$] as well. However, for equity markets (for example) the latter (the autocorrelation of absolute returns), is always found to be significantly positive). Add that to your tests!

The effect is called volatility clustering. Unusually large returns (of any sign) tend to be followed by unusually large returns (perhaps with the sign reversed). The net result is that financial researchers do *not* believe returns are i.i.d. -- even though most believe that markets are highly efficient.