Glosten-Milgrom Spread with Privacy Noise

This research derives closed-form bid-ask spreads for a classic market-making model under information-theoretic privacy constraints. It provides a formal framework for understanding how signal noise impacts liquidity pool transfers and informed trader fractions.

Strategy Decoder Editorial · 2026-05-19 · 5 min read

Key takeaways

Information-theoretic privacy mechanisms, specifically 'flip-noise,' directly impact bid-ask spreads in market-making models.
A 'privacy subsidy' represents a transfer of value from the market's liquidity pool to traders due to noisy information.
The research provides a closed-form solution for the bid-ask spread under these privacy conditions within the Glosten-Milgrom framework.
This work suggests implications for MPC-based matching engines that utilize differentially private trade direction signals.
The concept of a privacy subsidy demonstrates robustness across both continuous (Gaussian-Kyle) and discrete (Glosten-Milgrom) market models.

Market microstructure models are fundamental to understanding how prices are formed and how liquidity is provided in financial markets. A pivotal model in this field is the Glosten-Milgrom sequential-trading model, which describes how bids and asks are set by market makers interacting with informed and uninformed traders. Recent research by Yuki Nakamura, as reported on arXiv, introduces a novel angle to this classic framework by exploring the implications of information-theoretic privacy on bid-ask spreads and overall welfare.

Understanding the Glosten-Milgrom Model with Noise

The original Glosten-Milgrom model posits a market where a market maker, uncertain about the true value of an asset, faces a sequence of buy or sell orders. These orders come from either informed traders (who know the asset's true value) or uninformed liquidity traders. The market maker adjusts their bid and ask prices based on the order flow to avoid losses to informed traders. The bid-ask spread in this model reflects the compensation required by the market maker for the risk of trading against better-informed participants.

Nakamura's research extends this by introducing a "binary flip channel" to the market maker's observation of trade direction. This means that the signal indicating whether a trade is a buy or a sell is intentionally perturbed with a certain probability, denoted as $\eta$. This perturbation serves as an information-theoretic model for privacy mechanisms. Essentially, the market maker does not observe the true trade direction with 100% accuracy, but rather a noisy version of it.

The Derivation of the Bid-Ask Spread

Under a committed Bayesian pricing rule, where the market maker rationally updates their beliefs based on the noisy signal, the paper derives a closed-form expression for the equilibrium bid-ask spread. This spread is given by $\mu(1-2\eta)\Delta$, where:

$\mu$ represents the fraction of informed traders in the market.
$\eta$ is the probability of the trade direction signal being 'flipped' or perturbed by the privacy mechanism.
$\Delta = v_H - v_L$ is the range between the high and low possible asset values.

This formula reveals a direct relationship between the level of noise ($\eta$) and the bid-ask spread. As $\eta$ increases (more noise, more privacy), the term $(1-2\eta)$ decreases, suggesting a narrower spread. This is a subtle yet crucial point: increased noise makes it harder for the market maker to infer information from trades, which can lead to a reduction in the informational component of the spread if the noise is sufficiently high. However, it also has implications for the market maker's ability to accurately price risk.

The Concept of a "Privacy Subsidy"

One of the most intriguing findings from this research is the identification of a "privacy subsidy." This subsidy is described as a per-trade transfer of $\mu\eta\Delta$ from the protocol's liquidity pool directly to the traders. In simpler terms, when information is obfuscated through privacy mechanisms (represented by $\eta$), the market maker's compensation for adverse selection might decrease, effectively subsidizing traders at the expense of the liquidity providers.

This concept mirrors prior work on Gaussian-Kyle models, extending the notion of a privacy subsidy from continuous to discrete market microstructure settings. The robustness of this concept across different classical models (Gaussian-Kyle and Glosten-Milgrom) suggests a fundamental principle at play: intentionally introducing noise into market signals can shift value between market participants and liquidity providers.

Applications in Modern Market Design

The primary practical application highlighted by the paper is in the context of Multi-Party Computation (MPC)-based matching engines. These advanced systems aim to maintain trader privacy by disclosing trade direction signals in an $\varepsilon$-differentially-private manner. In such setups, the market maker (or the pricing algorithm) would base its pricing decisions on these deliberately noisy direction signals. Understanding how this noise impacts spreads and welfare is critical for designing fair and efficient MPC-based trading systems.

Differentially private mechanisms are becoming more prevalent in various data-sharing scenarios, and their application to financial market infrastructure is a logical—and complex—next step. Nakamura's work provides a theoretical underpinning for quantifying the costs and benefits of introducing such privacy measures in market operations.

Key Factors Influencing the Spread

The derived spread formula $\mu(1-2\eta)\Delta$ emphasizes several critical factors:

Informed Trader Fraction ($\mu$): A higher proportion of informed traders generally leads to a wider spread, as predicted by classic models. The market maker incurs greater risk of trading against those with superior information.
Noise Probability ($\eta$): The presence of privacy-inducing noise has a multiplicative effect. If $\eta$ is zero (no noise), the spread is simply $\mu\Delta$. As $\eta$ increases, the term $(1-2\eta)$ reduces, potentially narrowing the spread. This implies that while noise makes it harder to identify informed trades, it also changes the market maker's perception of risk.
Value Range ($\Delta$): A larger potential profit for informed traders (i.e., a wider range between possible asset values) necessitates a larger spread to compensate the market maker.

This framework allows for quantitative analysis of how different privacy levels (quantified by $\eta$) interact with the underlying market structure to determine liquidity costs.

Why it matters for algo traders

For algorithmic and quantitative traders, this research offers several crucial insights. Firstly, it provides a deeper theoretical understanding of how market microstructure reacts to information asymmetry when that information is intentionally obscured. If market venues or central clearing parties begin implementing privacy-preserving matching engines, the bid-ask spreads observed will be directly influenced by parameters like $\eta$. Algo traders need to incorporate such mechanisms into their models for accurate price prediction, optimal order placement, and effective liquidity provision.

Secondly, the concept of a "privacy subsidy" suggests that there might be inherent transfers of value embedded within these privacy-enhanced market structures. Quantifying this transfer can help in designing strategies that exploit or mitigate these shifts. For liquidity-providing algorithms, understanding how their effective spread is being impacted by privacy noise is paramount. For informed-trading algorithms, a narrower spread due to privacy mechanisms could imply higher profitability per trade, assuming they retain their information advantage.

Finally, this work can inform the development and backtesting of execution algorithms in environments where trade signals are partially masked. The formula for the bid-ask spread under flip-noise offers a tangible way to model and stress-test algorithmic strategies against different privacy settings, preparing them for the evolving landscape of transparent yet privacy-conscious financial markets.

Frequently asked questions

What is the Glosten-Milgrom model?

The Glosten-Milgrom model is a foundational market microstructure model describing how market makers set bid and ask prices based on order flow from informed and uninformed traders to cover losses from trading against those with superior information.

What is a 'privacy subsidy' in this context?

A 'privacy subsidy' refers to a transfer of value from the market's liquidity providers to traders when information about trade direction is made noisy or private, effectively reducing the market maker's compensation for adverse selection.

How does privacy-induced noise affect the bid-ask spread?

According to the research, a market maker observing trade direction perturbed by 'flip-noise' might see a reduced bid-ask spread. This is because increased noise makes it harder to extract information, leading to the market maker adjusting their risk premium based on the obfuscated signal.

Tags: market microstructure, market making, bid-ask spread, information theory

Based on reporting by arXiv q-fin.TR.