  The capitalization-weighted total relative variation $\sum_{i=1}^d
\int_0^\cdot \mu_i (t) \mathrm{d} \langle \log \mu_i \rangle (t)$ in an equity
market consisting of a fixed number $d$ of assets with capitalization weights
$\mu_i (\cdot)$ is an observable and nondecreasing function of time. If this
observable of the market is not just nondecreasing, but actually grows at a
rate which is bounded away from zero, then strong arbitrage can be constructed
relative to the market over sufficiently long time horizons. It has been an
open issue for more than ten years, whether such strong outperformance of the
market is possible also over arbitrary time horizons under the stated
condition. We show that this is not possible in general, thus settling this
long-open question. We also show that, under appropriate additional conditions,
outperformance over any time horizon indeed becomes possible, and exhibit
investment strategies that effect it.
