Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Equilibrium with Heterogeneous Information Flows

Scott Robertson

Abstract

We study a continuous time economy where throughout time, insiders receive private signals regarding the risky assets' terminal payoff. We prove existence of a partial communication equilibrium where, at each private signal time, the public receives a signal of the same form as the associated insider, but of lower quality. This causes a jump in both the public information flow and equilibrium asset price. The resultant markets, while complete between each jump time, are incomplete over each jump. After establishing equilibrium for a finite number of private signal times, we consider the limit as the private signals become more and more frequent. Under appropriate scaling we prove convergence of the public filtration to the natural filtration generated by both the fundamental factor process $X$ and a continuous time process $J$ taking the form $J_t = X_1 + Y_t$ where $X_1$ is the terminal payoff and $Y$ an independent Gaussian process. This coincides with the filtration considered in 'Additional Utility of Insiders with Imperfect Dynamical Information' (Corcuera, et al. Finance & Stochastics 2004). However, while therein the filtration was exogenously assumed to be that of an insider who observes a private signal flow, here it arises endogenously as the public filtration when there are a large number of insiders receiving signals throughout time.

Equilibrium with Heterogeneous Information Flows

Abstract

We study a continuous time economy where throughout time, insiders receive private signals regarding the risky assets' terminal payoff. We prove existence of a partial communication equilibrium where, at each private signal time, the public receives a signal of the same form as the associated insider, but of lower quality. This causes a jump in both the public information flow and equilibrium asset price. The resultant markets, while complete between each jump time, are incomplete over each jump. After establishing equilibrium for a finite number of private signal times, we consider the limit as the private signals become more and more frequent. Under appropriate scaling we prove convergence of the public filtration to the natural filtration generated by both the fundamental factor process and a continuous time process taking the form where is the terminal payoff and an independent Gaussian process. This coincides with the filtration considered in 'Additional Utility of Insiders with Imperfect Dynamical Information' (Corcuera, et al. Finance & Stochastics 2004). However, while therein the filtration was exogenously assumed to be that of an insider who observes a private signal flow, here it arises endogenously as the public filtration when there are a large number of insiders receiving signals throughout time.
Paper Structure (19 sections, 16 theorems, 205 equations, 2 figures)

This paper contains 19 sections, 16 theorems, 205 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Model
  3. Equilibrium Definition and the Main Result
  4. Two insiders
  5. Exponential-Gaussian Equilibrium with Common Information
  6. Continuous Time
  7. Single Period
  8. Conditional Densities and Connections with Signal Realizations
  9. Construction of the PCE
  10. An Example with Two Signal Times
  11. The Large Number of Insiders Limit
  12. The Effective Signal Process
  13. Convergence Analysis over $[0,1)$
  14. What Happens at $t=1$?
  15. On the Admissible Trading Strategies
  16. ...and 4 more sections

Key Result

Theorem 3.4

There exists a PCE $(\mathbb{F}^m,S)$. The market signals $\left\{H_n\right\}$ are from E:H_def. The filtration $\mathbb{F}^m$ is constructed as in E:filtnk_def, E:filtm_ik_def. Furthermore,

Figures (2)

  • Figure 1: From left to right, sample paths for the optimal trading strategies; the factor and price; and the market price of risk. Parameters are in \ref{['E:parameter']} and \ref{['E:parameter_1']}.
  • Figure 2: From left to right, insider welfare above the uninformed over $[t_2,1]$ as a function of the insider's arrival time $t_2$. Parameters are in \ref{['E:parameter']} along with $C_1=2,D_1=1$. The left plot has $C_2=D_2 = 1$; the middle plot has $C_2 = (1-t_2)^{-1}, D_2 = 1$; the right plot has $C_2 = D_2 = (1-t_2)^{-1}$.

Theorems & Definitions (39)

  • Remark 2.1
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Theorem 3.4
  • Proposition 4.2
  • Remark 4.3
  • Proposition 4.5
  • Remark 4.6
  • Lemma 5.1
  • ...and 29 more