  Trading styles can be classified into either trend-following or
mean-reverting. If the net trading style is trend-following the traded asset is
more likely to move in the same direction it moved previously (the opposite is
true if the net style is mean-reverting). The result of this is to introduce
positive (or negative) correlations into the time series. We here explore the
effect of these correlations on the long-run variance of the series through
probabilistic models designed to explicitly capture the direction of trading.
Our theoretical insights suggests that relative to random walk models of asset
prices the long-run variance is increased under trend-following strategies and
can actually be reduced under mean-reversal conditions. We apply these models
to some of the largest US stocks by market capitalisation as well as
high-frequency EUR/USD data and show that in both these settings, the ability
to predict the asset price is generally increased relative to a random walk.
