  We introduce a new formulation of asset trading games in continuous time in
the framework of the game-theoretic probability established by Shafer and Vovk
(Probability and Finance: It's Only a Game! (2001) Wiley). In our formulation,
the market moves continuously, but an investor trades in discrete times, which
can depend on the past path of the market. We prove that an investor can
essentially force that the asset price path behaves with the variation exponent
exactly equal to two. Our proof is based on embedding high-frequency
discrete-time games into the continuous-time game and the use of the Bayesian
strategy of Kumon, Takemura and Takeuchi (Stoch. Anal. Appl. 26 (2008)
1161--1180) for discrete-time coin-tossing games. We also show that the main
growth part of the investor's capital processes is clearly described by the
information quantities, which are derived from the Kullback--Leibler
information with respect to the empirical fluctuation of the asset price.
