Continuous-time Trading and the Emergence of Volatility

This note is part of the recent revival of interest in game-theoretic probability (see, e.g., [7, 8, 4, 2, 3]). It concentrates on the study of the“ √ dt effect”, the fact that a typical change in the value of a non-degenerate diffusion process over short time period dt has order of magnitude √ dt. Within the “standard” (not using non-standard analysis) framework of game-theoretic probability, this study was initiated in [9]. In our definitions, however, we will be following [11], which also establishes some other randomness-type properties of continuous price processes. The words such as “positive”, “negative”, “before”, and “after” will be understood in the wide sense of ≥ or ≤, respectively; when necessary, we will add the qualifier “strictly”.