This note continues investigation of randomness-type properties emerging in idealized financial markets with continuous price processes. It is shown, without making any probabilistic assumptions, that the strong variation exponent of non-constant price processes has to be 2, as in the case of continuous martingales.
"Continuous-time trading and the emergence of volatility." Electron. Commun. Probab. 13 319 - 324, 2008. https://doi.org/10.1214/ECP.v13-1383