A Game - Theoretic Explanation of the √ dt Effect

The returns from a traded security typically have the order of magnitude √ dt. This can be explained by the absence of riskless opportunities for making money: any different order of magnitude would permit such opportunities. The explanation has been made rigorous in the standard measure-theoretic framework for continuous-time probability, but this requires the assumption that the security’s price process follows a particular stochastic process (usually fractional Brownian motion). This note makes the explanation rigorous without any stochastic assumption, using our purely game-theoretic framework for finance [7]. This is a preliminary draft.