Beating the market? A mathematical puzzle for market efficiency

The efficient market hypothesis is highly discussed in economic literature. In its strongest form, it states that there are no price trends. When weakening the non-trending assumption to arbitrary short, small, and fully unknown trends, we mathematically prove for a specific class of control-based trading strategies positive expected gains. These strategies are model free, i.e., a trader neither has to think about predictable patterns nor has to estimate market parameters such as the trend’s sign like momentum traders have to do. That means, since the trader does not have to know any trend, even trends too small to find are enough to beat the market. Adjustments for risk and comparisons with buy-and-hold strategies do not satisfactorily solve the problem. In detail, we generalize results from the literature on control-based trading strategies to market settings without specific model assumptions, but with time-varying parameters in discrete and continuous time. We give closed-form formulae for the expected gain as well as the gain’s variance and generalize control-based trading rules to a setting where older information counts less. In addition, we perform an exemplary backtesting study taking transaction costs and bid-ask spreads into account and still observe—on average—positive gains.