Insider Trading and Nonlinear Equilibria: Single Auction Case

A nonlinear version of the Kyle [1985] model is studied. If linear structure might work for small orders, it would hardly be the case for large orders. No restriction is made neither on the form of equilibrium nor on the probability distribution of the ex-ante asset value. Equilibrium is characterized as a fixed point of an operator, which depends only on the distribution of the asset value; that is, the equilibrium is fully determined by the probability distribution of the asset value. A necessary and sufficient condition for the existence of the equilibrium is established. Furthermore, some explicit examples of equilibria are explored. In the simple case of Bernoulli distribution (just good news and bad news), it is shown that there is a unique equilibrium in which the price is strongly nonlinear and has plausible empirical counterparts. The problem is more complex if the ex-ante asset value is a continuous random variable. In this case, we restrict ourselves to a class of equilibria in which the price can be obtained explicitly. The existence of a unique equilibrium is then characterized in this class, and this provides Kyle's linear equilibrium as an example. The paper moves to the question on how risk aversion affects the equilibrium. Specifically, we assume that the insider has negative exponential utility, and prove that there exists a unique linear equilibrium, in which both quality of the signal and the initial position play important role. It is not surprising that the price pressure is lower than that in the risk neutral case. As far as the insider's strategy is concerned, insiders taking long-position (such as corporate insiders) want more to sell when the asset price is to go downwards than to buy when the price is to go upwards, and in this case, the price becomes higher than the risk averse price as long as the aggregated market order is smaller than some sufficiently large number. Naturally, the risk averse equilibrium converges uniformly to the risk neutral equilibrium as the risk aversion rate tends to zero.