Financial equilibrium with asymmetric information and random horizon

TL;DR

This paper explicitly solves a Kyle model with a random exponential trading horizon, analyzing equilibrium conditions, market maker signals, and the behavior of market depth, revealing new dynamics compared to the classical model.

Contribution

It provides explicit solutions for Kyle's model with a random horizon, characterizes equilibrium prices and strategies, and uncovers novel properties of market depth dynamics.

Findings

Equilibrium exists only if the final payoff is Bernoulli distributed.

Market maker signals are time-changed versions of the Bernoulli case.

Kyle's lambda is a uniformly integrable supermartingale, converging to zero in the Bernoulli case.

Abstract

We study in detail and explicitly solve the version of Kyle's model introduced in a specific case in \cite{BB}, where the trading horizon is given by an exponentially distributed random time. The first part of the paper is devoted to the analysis of time-homogeneous equilibria using tools from the theory of one-dimensional diffusions. It turns out that such an equilibrium is only possible if the final payoff is Bernoulli distributed as in \cite{BB}. We show in the second part that the signal of the market makers use in the general case is a time-changed version of the one that they would have used had the final payoff had a Bernoulli distribution. In both cases we characterise explicitly the equilibrium price process and the optimal strategy of the informed trader. Contrary to the original Kyle model it is found that the reciprocal of market's depth, i.e. Kyle's lambda, is a uniformly integrable supermartingale. While Kyle's lambda is a potential, i.e. converges to $0$, for the Bernoulli distributed final payoff, its limit in general is different than $0$.