Financial equilibrium with asymmetric information and random horizon

Umut Çetin

doi:10.1007/s00780-017-0348-0

Abstract

We study in detail and explicitly solve the version of Kyle’s model introduced in a specific case in Back and Baruch (Econometrica 72:433–465, 2004), 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 Back and Baruch (Econometrica 72:433–465, 2004). We show in the second part that the signal the market makers use in the general case is a time-changed version of the one 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. In contrast to the original Kyle model, it is found that the reciprocal of the 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 from 0.

Highlights

The canonical model of markets with asymmetric information is due to Kyle [13]
There are mainly three types of agents that constitute the market: a strategic risk-neutral informed trader with a private information regarding the future value of the asset, non-strategic uninformed noise traders, and a number of risk-neutral market makers competing for the net demand from the strategic and non-strategic traders
Using tools from potential theory for one-dimensional diffusions, we have solved a version of the Kyle model with general payoffs when the announcement date has an exponential distribution and is independent of all other parameters of the model

Summary

Introduction

The canonical model of markets with asymmetric information is due to Kyle [13]. Kyle studies a market for a single risky asset whose price is determined in equilib-. The finiteness of the time change implies that the limiting distribution of the market makers’ signal is a non-degenerate normal distribution This particular feature allows us to extend the model of Back and Baruch to a much more general setting, including the normally distributed case considered in [8]. On the other hand, from the earlier works on this subject is that the equilibrium prices exhibit a jump at the announcement date τ This is only natural since τ is unknown to the informed trader, and there is no strategy to ensure that the market price converges almost surely to Γ as time approaches τ , which is a totally inaccessible stopping time even for the informed.

The setup

There exists a unique strong solution to t

Bernoulli-distributed liquidation value

More general liquidation value

Conclusion