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Solvability of the Gaussian Kyle model with imperfect information and risk aversion

Reda Chhaibi, Ibrahim Ekren, Eunjung Noh

TL;DR

We address solvability of a Gaussian Kyle model with a risk-averse insider who has imperfect information about the terminal price. By combining an optimal-transport formulation with a filtering problem under a historical measure, we construct an equilibrium where the insider controls both a path-dependent state and the conditional distribution of the price given market-maker information. The analysis yields explicit expressions for Kyle’s lambda and the pricing rule in Gaussian settings, with a measure change to ${\\mathbb Q}$ facilitating tractable optimization. The results extend classical Kyle theory to risk-averse, partially informed insiders and reveal how information quality shapes price impact and market-maker compensation. The approach provides a solvable framework that links transport maps, Gaussian filtering, and market microstructure in continuous time.

Abstract

We investigate a Kyle model under Gaussian assumptions where a risk-averse informed trader has imperfect information on the fundamental price of an asset. We show that an equilibrium can be constructed by considering an optimal transport problem that is solved under a measure that renders the utility of the informed trader martingale and a filtering problem under the historical measure.

Solvability of the Gaussian Kyle model with imperfect information and risk aversion

TL;DR

We address solvability of a Gaussian Kyle model with a risk-averse insider who has imperfect information about the terminal price. By combining an optimal-transport formulation with a filtering problem under a historical measure, we construct an equilibrium where the insider controls both a path-dependent state and the conditional distribution of the price given market-maker information. The analysis yields explicit expressions for Kyle’s lambda and the pricing rule in Gaussian settings, with a measure change to facilitating tractable optimization. The results extend classical Kyle theory to risk-averse, partially informed insiders and reveal how information quality shapes price impact and market-maker compensation. The approach provides a solvable framework that links transport maps, Gaussian filtering, and market microstructure in continuous time.

Abstract

We investigate a Kyle model under Gaussian assumptions where a risk-averse informed trader has imperfect information on the fundamental price of an asset. We show that an equilibrium can be constructed by considering an optimal transport problem that is solved under a measure that renders the utility of the informed trader martingale and a filtering problem under the historical measure.

Paper Structure

This paper contains 14 sections, 8 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Model
  4. Statement of the problem
  5. Main result
  6. Construction of an equilibrium
  7. Properties of the equilibrium
  8. Necessary conditions on Markovian equilibria
  9. Optimal transport
  10. Proofs
  11. Proof of Lemma \ref{['lem:v']}
  12. Proof of Lemma \ref{['lem:sigma']}
  13. Proof of Theorem \ref{['thm:main']}
  14. Proof of Theorem \ref{['thm:prop']}

Key Result

Lemma 3.1

All other parameters being positive, for all $\varepsilon,\gamma\geq 0$ there exists a unique positive solution $v = v\left( \varepsilon, \gamma, \sigma^2 T, \sigma_\Xi^2, \sigma_\beta^2 \right) > 0$ to the equation satisfying

Theorems & Definitions (19)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 3.1
  • Remark 3.2
  • Lemma 3.3
  • Theorem 3.4: Existence of Equilibrium
  • Remark 3.5
  • Theorem 3.6: Properties of the Equilibrium
  • Remark 3.7
  • ...and 9 more