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A continuous-time Kyle model with price-responsive traders

Eunjung Noh

TL;DR

This paper addresses how price-responsive noise traders alter price discovery in a continuous-time Kyle framework. It develops a linear–Gaussian model where momentum and contrarian traders respond to price innovations, and shows that a finite-dimensional Kalman–Bucy filter governs price inference, yielding a forward–backward Riccati system that couples insider optimization with filtering. The authors establish local existence and uniqueness of equilibrium under weak feedback and derive first-order comparative statics showing that price informativeness rises while insider profits fall as feedback strengthens. They also identify three pathways to breakdown under strong feedback: Riccati blow-up, loss of contraction of the equilibrium map, and instability of the Kalman filter, which can yield multiple equilibria and destabilize price discovery. Overall, the framework extends the Kyle model to endogenous feedback while retaining analytical tractability, enabling analysis of learning, information aggregation, and stability in markets with price-reactive trading.

Abstract

Classical Kyle-type models of informed trading typically treat noise trader demand as purely exogenous. In reality, many market participants react to price movements and news, generating feedback effects that can significantly alter market dynamics. This paper develops a continuous-time Kyle framework in which two types of price-responsive traders (momentum and contrarian traders) adjust their demand in response to price signals. This extension yields a finite-dimensional Kalman filter for price discovery and leads to a forward-backward Riccati system characterizing equilibrium. We show that when feedback is weak, equilibrium exists and is unique as a smooth perturbation of the classical Kyle solution, allowing us to derive explicit comparative statics for insider profits and price informativeness. For stronger feedback, the model generates rich dynamics, including potential multiplicity of equilibria and amplification effects. Our framework thus bridges the gap between purely exogenous noise and more realistic, behaviorally motivated trading.

A continuous-time Kyle model with price-responsive traders

TL;DR

This paper addresses how price-responsive noise traders alter price discovery in a continuous-time Kyle framework. It develops a linear–Gaussian model where momentum and contrarian traders respond to price innovations, and shows that a finite-dimensional Kalman–Bucy filter governs price inference, yielding a forward–backward Riccati system that couples insider optimization with filtering. The authors establish local existence and uniqueness of equilibrium under weak feedback and derive first-order comparative statics showing that price informativeness rises while insider profits fall as feedback strengthens. They also identify three pathways to breakdown under strong feedback: Riccati blow-up, loss of contraction of the equilibrium map, and instability of the Kalman filter, which can yield multiple equilibria and destabilize price discovery. Overall, the framework extends the Kyle model to endogenous feedback while retaining analytical tractability, enabling analysis of learning, information aggregation, and stability in markets with price-reactive trading.

Abstract

Classical Kyle-type models of informed trading typically treat noise trader demand as purely exogenous. In reality, many market participants react to price movements and news, generating feedback effects that can significantly alter market dynamics. This paper develops a continuous-time Kyle framework in which two types of price-responsive traders (momentum and contrarian traders) adjust their demand in response to price signals. This extension yields a finite-dimensional Kalman filter for price discovery and leads to a forward-backward Riccati system characterizing equilibrium. We show that when feedback is weak, equilibrium exists and is unique as a smooth perturbation of the classical Kyle solution, allowing us to derive explicit comparative statics for insider profits and price informativeness. For stronger feedback, the model generates rich dynamics, including potential multiplicity of equilibria and amplification effects. Our framework thus bridges the gap between purely exogenous noise and more realistic, behaviorally motivated trading.
Paper Structure (22 sections, 8 theorems, 87 equations)

This paper contains 22 sections, 8 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Literature review
  3. Model
  4. Equilibrium Analysis
  5. State-space representation and price filtering
  6. Equilibrium restriction and inconspicuous trading
  7. Informed trader's optimization problem
  8. Covariance dynamics and Riccati system
  9. Weak Feedback
  10. Local existence and uniqueness of equilibrium
  11. Sufficient stability condition
  12. First-order comparative statics for small feedback
  13. Strong Feedback and Breakdown of Equilibrium
  14. Loss of global existence of the Riccati equation
  15. Loss of uniqueness of the equilibrium fixed point
  16. ...and 7 more sections

Key Result

Lemma 4.1

(Inconspicuousness of linear strategies) Suppose the informed trader follows a linear strategy of the form where $\beta_t$ is deterministic and locally square-integrable. Then, under rational pricing, the informed trader’s order flow is inconspicuous in the sense that

Theorems & Definitions (26)

  • Remark 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Proposition 4.3: Well-posedness of the covariance Riccati system
  • proof
  • Remark 4.4: Well-posedness and Economic Interpretation
  • Theorem 5.1
  • ...and 16 more