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Strategic Learning and Trading in Broker-Mediated Markets

Alif Aqsha, Fayçal Drissi, Leandro Sánchez-Betancourt

TL;DR

This paper analyzes strategic learning in a broker-mediated market where both the broker and an informed trader filter hidden information. By employing Kalman-Bucy filtering and linear-quadratic control, it derives explicit Markovian strategies for both agents under two information regimes: learning from prices and learning from trading flow. The key finding is that information leakage via order flow confers a substantial and cost-parity economic value to the broker, while relying on prices alone yields a performance close to a naive, noise-internalising strategy. The results imply a tangible advantage for brokers who have access to client flow, with meaningful implications for privacy, execution, and market efficiency.

Abstract

We study strategic interactions in a broker-mediated market in which agents learn and exploit each other's private information. A broker provides liquidity to an informed trader and to noise traders while managing inventory in a lit market. The informed trader infers the broker's trading activity in the lit market, while the broker estimates the trader's private signal. Information leakage in the client's trading flow generates economic value for the broker that is comparable in magnitude to transaction costs: the broker can speculate profitably and manage risk more effectively, which in turn adversely affects the informed trader's performance. Brokers therefore hold a strategic advantage over traders who rely solely on prices to filter information. When the broker only relies on prices rather than client trading flow to infer information, their trading performance becomes indistinguishable from the performance of a naive strategy that internalises noise flow, externalises informed flow, and offloads inventory at a constant rate.

Strategic Learning and Trading in Broker-Mediated Markets

TL;DR

This paper analyzes strategic learning in a broker-mediated market where both the broker and an informed trader filter hidden information. By employing Kalman-Bucy filtering and linear-quadratic control, it derives explicit Markovian strategies for both agents under two information regimes: learning from prices and learning from trading flow. The key finding is that information leakage via order flow confers a substantial and cost-parity economic value to the broker, while relying on prices alone yields a performance close to a naive, noise-internalising strategy. The results imply a tangible advantage for brokers who have access to client flow, with meaningful implications for privacy, execution, and market efficiency.

Abstract

We study strategic interactions in a broker-mediated market in which agents learn and exploit each other's private information. A broker provides liquidity to an informed trader and to noise traders while managing inventory in a lit market. The informed trader infers the broker's trading activity in the lit market, while the broker estimates the trader's private signal. Information leakage in the client's trading flow generates economic value for the broker that is comparable in magnitude to transaction costs: the broker can speculate profitably and manage risk more effectively, which in turn adversely affects the informed trader's performance. Brokers therefore hold a strategic advantage over traders who rely solely on prices to filter information. When the broker only relies on prices rather than client trading flow to infer information, their trading performance becomes indistinguishable from the performance of a naive strategy that internalises noise flow, externalises informed flow, and offloads inventory at a constant rate.
Paper Structure (37 sections, 8 theorems, 114 equations, 10 figures, 3 tables)

This paper contains 37 sections, 8 theorems, 114 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Literature review
  3. Modelling framework
  4. The informed trader
  5. Learning equations
  6. The control problem of the informed trader
  7. The optimal adaptive strategy
  8. The broker
  9. Learning equations
  10. Learning from prices
  11. Learning from trading flow
  12. The control problems of the broker
  13. Solving the control problem when the broker filters prices
  14. Solving the control problem when the broker filters the trading speed
  15. Numerical results
  16. ...and 22 more sections

Key Result

Theorem 1

The HJB equation eq:trader_hjb admits a unique solution in $C^{1,2}([0,T], \mathbb{R}^{|\mathfrak{x}|})$ in the form of where $g_i^I$, $i=0,1,2$ are defined in the appendix (Lemma lemma:trader_DEs_existence_solutions). Moreover, there exists a constant $c<\infty$ such that Finally, the trader's Markovian optimal trading rate for the control problem eq:trader_control_problem in feedback form is

Figures (10)

  • Figure 1: One sample path and $90\%$ confidence bands across simulations for the following processes. The broker's trading speed $\nu_t$, the trading speed $\eta_t$ of the informed trader, the cash process $X^{I}$ of the informed trader, the cash process $X^{B}$ of the broker, the inventory $Q^{I}$ of the informed trader, and the inventory $Q^{B}$ of the broker. The top panel is when the broker filters from prices and the bottom panel is when she filters from the informed trader's flow.
  • Figure 2: One realisation of filtering carried by both informed trader and broker. The top panel is when the broker filters from prices and the bottom panel is when she filters from the informed trader's flow.
  • Figure 3: The additive components of the broker's strategy. The function $h_{(\cdot)}^B$ refers to the coefficient of component $(\cdot)$ in the strategy; the superscript "${\textup{alt}}$" refers to the second strategy with the alternative filter. The first row is showing the four additive components when the broker filters from prices. The second and third row are showing the five additive components when the broker filters from the informed trader's flow.
  • Figure 4: One sample path for the signal $\alpha$ and the new estimates $\hat{\alpha}$ when the broker mispecifies the informed trader's inventory.
  • Figure 5: The components of the informed trader's strategy.
  • ...and 5 more figures

Theorems & Definitions (17)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Proposition 1
  • Proposition 2
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • proof
  • ...and 7 more