Equilibria in Kyle games : existence and construction

In the first part of this thesis, we present a discrete time version of Kyle’s (1985) classic model of insider trading. The model has three kinds of traders: an insider, random noise traders, and a market maker. The insider aims to exploit her informational advantage and maximise expected profits while the market maker observes the total order flow and sets prices accordingly. We show how the multi-period model with finitely many pure strategies can be reduced to a (static) social system in the sense of Debreu (1952) and prove the existence of a sequential Kyle equilibrium, following Kreps and Wilson (1982). This requires no probabilistic restrictions on the true value, the insider’s dynamic information, and the noise trader’s actions. In the single-period model we establish bounds for the insider’s strategy in equilibrium. In addition, we prove the existence of an equilibrium for the game with a continuum of actions, by considering an approximating sequence of games with finitely many actions. Because of the lack of compactness of the set of measurable price functions, standard infinite-dimensional fixed point theorems are not applicable. The second part of this thesis is concerned with the Binomial Kyle model, a discrete, simplified approach to the continuous model of Back (1992). The noise trader’s demand is given by a simple symmetric random walk and the true value has binomial probabilities. We study inconspicuous Kyle equilibria, a refinement of the Kyle equilibrium concept, in which the total demand of the insider and noise trader has the same distribution as the noise trader’s demand alone. We explicitly construct an inconspicuous Kyle equilibrium in the case of a binary true value. The total demand process of the insider in equilibrium possesses a bridge structure similar to the one in the Poisson process model of Çetin and Xing (2013), and can be considered a purely discrete counterpart to the Markov bridges in Çetin and Danilova (2018). We show that there exists no inconspicuous trade equilibrium when the number of true values is maximal.