The appearance of a Brownian term in the price dynamics on a stock market was interpreted in [De Meyer, Moussa-Saley (2003)] as a consequence of the informational asymmetries between agents. To take beneﬁt of their private information without revealing it to fast, the informed agents have to introduce a noise on their actions, and all these noises introduced in the day after day transactions for strategic reasons will aggregate in a Brownian Motion. We prove in the present paper that this kind of argument leads not only to the appearance of the Brownian motion, but it also narrows the class of the price dynamics: the price process will be, as deﬁned in this paper, a continuous martingale of maximal variation. This class of dynamics contains in particular Black and Scholes’ as well as Bachelier’s dynamics. The main result in this paper is that this class is quite universal and independent of a particular model: the informed agent can choose the speed of revelation of his private information. He determines in this way the posterior martingale L , where L q is the expected value of an asset at stage q given the information of the uninformed agents. The payoﬀ of the informed agent at stage q can typically be expressed as a 1-homogeneous function M of L q + 1 – L q . In a game with n stages, the informed agent will therefore chose the martingale Ln that maximizes the M -variation. Under a mere continuity hypothesis on M , we prove in this paper that Ln will converge to a continuous martingale of maximal variation. This limit is independent of M .
De Meyer, Bernard, "Price Dynamics on a Stock Market with Asymmetric Information" (2007). Cowles Foundation Discussion Papers. 1897.