Information in Securities Markets: Kyle Meets Glosten and Milgrom

We study a model of informed trading in which trades arrive sequentially, uninformed trades arriving as a Poisson process. We characterize an equilibrium in which the single informed trader plays a mixed strategy – a point process with stochastic intensity. In this equilibrium, informed and uninformed trades arrive probabilistically, as they are assumed to do in Glosten-Milgrom models. We study a sequence of such markets in which uninformed trades become smaller and arrive more frequently, approximating a Brownian motion. We show that if the equilibria converge then their limit is the equilibrium of a Kyle-type continuous auctions model. ∗John M. Olin School of Business, Washington University in St. Louis, St. Louis, MO 63130, [email protected]. †David Eccles School of Business, University of Utah, Salt Lake City, UT 84112, [email protected]. The three principal models of asymmetric information in securities markets are described in Grossman (1981), Kyle (1985) and Glosten-Milgrom (1985). Models of the last two types have been used extensively in recent years to explore various issues in finance. The Kyle model is an auction or batch-trading model. Traders submit demands and then market makers compete in a Bertrand fashion to supply the net order. In the GlostenMilgrom model, orders arrive individually: market makers compete, again in a Bertrand way, to quote bid/ask prices at which orders are transacted. Very few securities markets are actually organized as periodic auctions. The justification for Kyle’s assumption is tractability. The Glosten-Milgrom model is relatively intractable if one wants to solve for a full equilibrium in which informed traders optimize their times of trade, and it has heretofore been assumed that the arrival of informed and uninformed traders is determined by some exogenous process. O’Hara (1995, p. 59) describes this aspect of the Glosten-Milgrom model well: How traders actually arrive at the market is an important issue. Informed traders profit from trading if prices are not at full-information levels, and so any informed trader will prefer to trade as much (and as often) as possible. Since such behavior would quickly indicate the information of the informed, the market maker would quickly (perhaps instantly) adjust prices to reflect this information . . . One way to avoid this instantaneous revelation outcome is to assume that traders are chosen probabilistically . . . In this paper, we will develop a simple model that is a hybrid of the Kyle and Glosten-Milgrom models. The goal is to understand the significance of the limitations described above. How much damage does it do to assume markets are organized as auctions, or, on the other hand, to ignore the optimization of informed traders? In our model, uninformed liquidity-motivated trades arrive as a Poisson process. We show that an informed trader will follow a mixed strategy in equilibrium, so informed and uninformed trades do arrive probabilistically. We then consider the limit of a sequence of such markets, in which the uninformed trades are smaller and arrive more frequently, the Poisson process converging to a Brownian motion. Kyle derives a continuous-time model as the limit of a sequence of periodic auctions. Our main result is that the limits of the periodic auction model and the sequential