Optimal Quoting under Adverse Selection and Price Reading

Alexander Barzykin; Malo Lemmel; Olivier Gu\'eant; Philippe Bergault

REVIEW 3 major objections 4 minor 4 cited by

Optimal Quoting under Adverse Selection and Price Reading

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A first-order formula gives market makers explicit quote corrections for informed flow and skew sniffers, split into global and tier-specific parts.

desk verdict Useful, honest extension of optimal market making to adverse selection and price reading, but the ε=1 simulations stretch the first-order perturbation further than the paper can guarantee. read the letter →

arxiv 2508.20225 v6 pith:KD5WQQNE submitted 2025-08-27 q-fin.TR q-fin.RM

Alexander Barzykin , Philippe Bergault , Olivier Guéant , Malo Lemmel This is my paper · ORCID

classification q-fin.TRq-fin.RM MSC 91G8093E20

keywords marketmakingadverseselectionpricereadinginformationalriskstochasticoptimalcontrolperturbationanalysisquoteskewclienttiering

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to give market makers a tractable way to adjust their bid and ask quotes when two real-world risks are present: adverse selection (trading against better-informed clients) and price reading (clients inferring the dealer's inventory direction from the skew in their quotes). Working at first order in a small perturbation parameter, the authors derive explicit formulas—equations (12) and (13)—for the optimal quote corrections. The correction splits into a global component, which reflects the overall increase in informational risk and applies to every client tier, and a tier-specific component, which adjusts the skew shown to a particular tier to limit information leakage. The paper argues the resulting 'Optimal' quoting strategy beats both ignoring the risks and naively removing all skew in simulations, in terms of risk-adjusted PnL. If right, this turns two practitioner concerns that were mostly treated in static toy models into implementable, dynamic quoting adjustments.

What carries the argument

The engine of the paper is a first-order Taylor expansion of the HJB equation and its maximizers around the baseline market making model with no informational risk (ε=0). The load-bearing objects are the value-function correction f (solving the linear system (7)) and the tier-specific corrections g, which combine into equations (12)–(13) for the optimal quote offsets; the finite-difference operators D± and the concavity parameter c(δ) from the intensity functions govern the size of the adjustment. The quadratic Hamiltonian approximation of Section 4 turns the HJB into a Riccati equation whose solution yields f in closed form.

What would settle it

Solve the full HJB equation (6) numerically for the exponential functional forms of Section 5 using a convergent method, then compare the true optimal quotes and PnL with the first-order approximations (12)-(13) at ε=1 across the stated inventory range; a material divergence, or a PnL gap that reverses the ranking against the 'No Skew' strategy, would refute the paper's practical claim.

Watch

Extended reading notes

Core claim

The central claim is that, in an infinite-horizon optimal market making model, the first-order impact of adverse selection and price reading on optimal quotes can be computed explicitly and separates cleanly. The optimal bid and ask offsets in feedback form are the no-informational-risk quotes plus a correction equal to (1/c) times a bracket containing: (i) the finite difference of a value-function correction f (which aggregates the impact of all tiers' informational risk on expected PnL), and (ii) a tier-specific term g that depends on the slope of the price-impact function ζ, the intensity function Λ, and the skew-reading sensitivity J'. Under a quadratic approximation of the Hamiltonians,

Load-bearing premise

The paper's first-order expansions are derived under a small perturbation parameter ε, but the numerical experiments set ε=1 and are forced to impose a floor at zero on the adjusted quotes because in extreme regimes the approximation is questionable; if the omitted o(ε) terms are not small over the relevant inventory and quote ranges, equations (12)–(13) do not describe the true optimal quotes and the simulated gains vanish.

Editorial extensions

If this is right

Market makers can compute quote adjustments for each client tier with only baseline quotes and the solution of a linear system, no full nonlinear HJB solve needed.
Price reading leads to a two-sided response: spread widening at zero inventory plus reduced skew to tiers with skew sniffers, with the de-skewing strongest for infrequent, sensitive readers.
Adverse selection can be exploited: a tier trading on slow signals can be used as a 'signal subscription' where attractive top-of-book prices buy information, improving franchise risk management.
The formulas give a directional answer to whether to lean against or go with the skew: the net effect depends on relative slopes of intensity and price-impact functions, summarized by sign differences β−κ in exponential examples.
The infinite-horizon, first-order approach avoids the curse of dimensionality and extends naturally to multi-asset settings via the same quadratic approximation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the first-order approximations hold, the same decomposition—global value-function effect plus tier-specific control effect—may apply to other informational frictions (e.g., signals from order flow toxicity, latency advantages), giving a template for quote corrections beyond these two risks.
The 'signal subscription' interpretation suggests a testable strategy: deliberately quote tight to an informed tier at small sizes to extract directional information, and measure whether the value of that information exceeds the adverse selection cost—this could be validated with client-level flow data.
The authors' simulation sets ε=1 despite formal small-ε assumptions; a natural stress test is to compare equations (12)-(13) against a fully numerical HJB solution for realistic parameters, which would show the regime of validity and whether the floor at zero quotes is often binding.
The tier-specific de-skewing result predicts that in markets with a large share of skew sniffers, top-of-book skew should be lower than the no-information-leakage baseline; this is observable in dealer-to-client quote data, e.g., comparing quote ladders across platforms with different information transparency.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper does something real: it puts price reading (skew sniffing) and adverse selection into a dynamic optimal-control market-making model and derives first-order quote adjustments with a clean decomposition into a global value-function component and a tier-specific component. Equations (12)–(13) are not in the prior literature, and for the exponential case the formulas are explicit and check out. The paper is also unusually honest in the text: footnote 9 admits the first-order approximation is questionable in extreme situations, and the authors flag the need for a quote floor. That honesty matters.

The main soft spot is exactly there. The expansion parameter ε is small by construction, but every simulation runs at ε=1, where the corrections are as large as the baseline quotes themselves. Figures 1 and 2 show quote adjustments around 1bp against a top-of-book half-spread of roughly 0.33bp. So the regime tested is not a perturbative one. The paper gives no uniform bound on the o(ε) remainder, so the simulated "Optimal" PnL gains show only that this particular first-order heuristic beats two other heuristics in the same model—not that it approximates the true optimal control. That is a genuine limitation, not a fatal one. The central framework and formulas can still be right; the numerical evidence just does not carry the weight the conclusion puts on it.

The other soft spots are minor by comparison. The PnL comparisons are in-sample: the strategy is derived from the same model that generates the simulated flows, so the improvements are partly by construction. No code or data is released, though the parameter sets are described well enough to reproduce. The authors' own quadratic-Hamiltonian machinery from [6] is reused; that is fine because the new informational terms enter as generic inputs, not fitted quantities.

Should this go to peer review? Yes. The formal derivation is a real contribution, the limitations are stated in the paper, and the weaknesses are addressable through a full numerical solution at ε=1, a remainder check, or a different benchmark strategy. A serious referee could help the authors tighten the claim. I would cite it if I worked on dealer quoting; I'd also bring it to a reading group, but I'd say in advance that the simulations are suggestive, not conclusive.

So: send to referee, with a clear request to examine the ε=1 validity and to ask for either error control or a full numerical comparison.

Referee Report

3 major / 4 minor

Summary. The paper proposes an infinite-horizon market-making model, in the spirit of Cartea–Jaimungal/Guéant, in which the reference price is affected by (i) adverse selection through tier- and size-dependent price-impact functions ζ and (ii) price reading through a feedback term J applied to the weighted skew of the market maker's own bid/ask ladder. The authors introduce a perturbation parameter ε and derive a first-order expansion of the value function around the no-informational-risk baseline, with the correction f characterized by the linear system (7). They then obtain first-order expansions of the optimal bid/ask offsets, Eqs. (12)–(13), showing a decomposition into a global value-function component common to all tiers and a tier-specific component. In a quadratic-Hamiltonian approximation with symmetric bid/ask functions and ρ→0, the adjustments become closed-form. Numerical examples with exponential intensities and two client tiers illustrate the effects and report simulated PnL, risk, and risk-adjusted performance for 'No Action', 'No Skew', and 'Optimal' strategies.

Significance. If the first-order expansions are reliable, the paper delivers a practical and implementable extension of existing market-making models: the explicit formulas (12)–(13) are straightforward to evaluate from baseline quotes and the auxiliary function f, and the global/tier-specific decomposition provides actionable intuition for flow toxicity and skew-sniffing. A notable strength is that no parameter is fitted to data, so the adjustment formulas do not reduce to fitted quantities; the exponential case is worked out in closed form and the simulation study uses 10^5 paths. The main limitation is that the numerical section runs the perturbation at ε=1, where no error control is provided and the corrections are comparable to the baseline quotes. Thus the practical 'optimality' claim is under-supported, although the analytical machinery is plausible and coherent.

major comments (3)

[§5.2–5.3, footnote 9, Eqs. (12)–(13)] Eqs. (12)–(13) are first-order expansions in ε, but the simulations set ε=1, and footnote 9 admits a 0-floor is needed because 'the applicability of the first-order approximations is questionable' in extreme situations. With κ=3 bp^-1, the baseline half-spread is approximately 1/κ≈0.33 bp, while Figs. 1–2 show quote adjustments of order ±1 bp, so the corrections are not small relative to the unperturbed quotes. No uniform bound on the o(ε) remainder is supplied. The simulated PnL improvements of 'Optimal' over No Action/No Skew therefore do not establish that (12)–(13) approximate the true optimal controls; they only show that one heuristic beats two other heuristics inside the same model. The authors should provide a remainder estimate or a small-ε/numerical-HJB convergence check, or explicitly weaken the optimality claim.
[§3.3.2–§3.3.3, Eq. (7), Eq. (6)] The perturbation ansatz ϑ=θ+εf+o(ε) and the first-order expansion of the maximizers in §3.3.3 require differentiability of the value function and of the argmax with respect to ε at ε=0; these properties are assumed without proof. Equation (7) for f is obtained by formally differentiating the HJB equation (6), so a verification argument or explicit regularity assumptions are needed to ensure that the formal expansion actually corresponds to the true value function and optimal quotes. At minimum, the regularity conditions should be stated, and one benchmark case should be checked against a high-accuracy numerical solution of (6).
[§5.2–§5.3, Figs. 3–9] The PnL simulations are entirely in-sample and reuse the same model and the same first-order approximations. 'Optimal' is compared only with the 'No Action' and 'No Skew' heuristics, not with a policy obtained from a high-accuracy solution of the original HJB equation (6) or with a small-ε reference solution. Consequently, the simulations cannot distinguish 'closer to optimal' from 'different but lucky'. The reported improvements are internal-consistency checks of the first-order heuristic, not validation of the expansion's optimality.

minor comments (4)

[§5.3, Figs. 7–9] The horizontal axes in Figs. 7–9 are labeled '0.10% ... 0.90%', while the text and Figs. 3–5 use '10% ... 90%'. Clarify whether SVS is expressed as a percentage or a decimal fraction.
[Eq. (7) and §3.3.2] The Feynman-Kac representation for f requires integrability and growth conditions on f and on the intensities Λ; these conditions are not stated. Adding them would make the derivation complete.
[§4.2, Eqs. (9)–(10)] The notation O(ε^2) in equations for Aε and Bε can be read as an exact residual of the quadratic Hamiltonian approximation. It would be clearer to state explicitly that Aε and Bε are themselves expanded to first order in ε, so that (9)–(10) are only first-order equations.
[§5.1] The time units deserve clarification: Λ0 is given in day^-1 while the simulation horizon is T=10^4 s. The conversion between these units should be stated explicitly for reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the quote adjustments are derived as first-order expansions from the model primitives, not relabeled fits or self-citation imports.

full rationale

The paper's central equations (12)–(13) are obtained by a genuine perturbation argument: they expand the HJB value function as ϑ = θ + εf + o(ε) and expand the Hamiltonian maximizers in ε around the no-information baseline. The correction f solves the linear equation (7) by collecting O(ε) terms, and the quote corrections follow from the first-order condition of the Hamiltonian. No parameter is fitted to data, and no empirical quantity is renamed as a prediction. The only overlapping-author citation, [6], is used in Section 4 as a stated quadratic-Hamiltonian approximation method, not as an imported uniqueness or existence theorem, and Section 3's first-order derivation is independent of it. The numerical section simulates the same model to compare strategies, which is a self-consistency check rather than an independent empirical validation, but it is not a circular derivation. Concerns about setting ε=1 and about the lack of remainder bounds (footnote 9) are legitimate approximation-error and correctness issues, not instances of circularity.

Assumptions & free parameters 5 free parameters · 6 assumptions · 1 invented entities

The core derivation is self-contained given the standard market-making assumptions. The numerical claims depend on hand-chosen parameter values and the price-reading feedback assumption. No data are fitted, so there is no parameter-fitting circularity, but the simulation validation is in-sample and the expansion is formal.

free parameters (5)

Lambda_0^{n,k} baseline trade intensity levels
Hand-picked in Section 5.2 (e.g., total top-of-book intensity 2000 day^-1) for the simulations; not fitted to data. The central derivation treats them as generic model inputs.
kappa^{n,k} exponential intensity and impact decay
Set to 3 bp^-1 in Section 5.2; determines the baseline spread and the sign of adverse-selection effects through beta-kappa.
alpha^{n,k}, beta^{n,k} adverse selection amplitude and slope
Chosen as alpha_{2,k}=0.05 and beta_{2,k}=2 for the adverse-selection illustration in Section 5.3. The qualitative effects depend on whether beta is below or above kappa.
w^{n,k} price-reading weights
Chosen as w_{2,k}=exp(-sqrt(Delta_k))/1000 in Section 5.2 to represent skew sniffers. The central derivation treats the weights as inputs.
gamma, sigma, rho risk aversion, volatility, discount rate
Set to gamma=1e-4 bp^-1 M^-1, sigma=100 bp/day, and rho to 0 in the symmetric closed-form analysis. These are standard model parameters, not estimated from data.

assumptions (6)

domain assumption Trade arrivals are marked point processes with intensities Lambda^{n,k,b/a}(delta) depending only on the market maker's own quote offset relative to its reference price S_t.
Section 2.1, Eq. (3). Standard in Avellaneda-Stoikov and Cartea-Jaimungal style models.
domain assumption The reference price S_t reacts to informed trades via zeta^{n,k}(delta) and to quote asymmetry via J_n(weighted skew), and these reactions enter PnL through q dS.
Section 2.1, Eq. (1), and Remark 2. This is the paper's model of adverse selection and price reading.
domain assumption The optimization criterion is the infinite-horizon discounted risk-adjusted PnL (Eq. 5), and the associated HJB equation has a smooth solution with well-defined finite differences.
Section 2.2 and Eq. (6). Standard for this literature, but not proved here.
ad hoc to paper The perturbation expansion theta = theta_0 + epsilon f + o(epsilon) is valid, and the maximizers of the perturbed Hamiltonian are differentiable in epsilon at epsilon=0.
Section 3.3.2-3.3.3. No uniform error bound is proved; this premise is needed for formulas (7), (12), and (13).
ad hoc to paper The Hamiltonian can be replaced by its second-order Taylor expansion around (p=0, epsilon=0), yielding a quadratic value function.
Section 4.1-4.2, method from [6]. The closed-form quote adjustments (12)-(13) rely on this approximation.
ad hoc to paper For the closed-form insights, bid-ask symmetry and the limit rho to 0 are assumed.
Section 4.4. The general nonsymmetric case is not analyzed in closed form.

invented entities (1)

Price-reading feedback channel J_n(weighted average quote skew) in the reference price dynamics
purpose: Models skew sniffers inferring the market maker's inventory from the quote ladder and moving the price against the market maker.
Introduced in Eq. (1) and motivated by the practitioner article [12] and order-book predictability papers [16,18], but the paper provides no empirical estimate or falsifiable prediction outside its own simulations.

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Cite this review

Pith. "Pith review of Optimal Quoting under Adverse Selection and Price Reading." pith.science (2026). https://pith.science/paper/KD5WQQNE

@misc{pith2026250820225,
  author       = {Pith},
  title        = {Pith review of: Optimal Quoting under Adverse Selection and Price Reading},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KD5WQQNE}},
  note         = {Machine review of arXiv:2508.20225}
}

read the original abstract

Over the past decade, many dealers have implemented algorithmic models to automatically respond to RFQs and manage flows originating from electronic platforms. In parallel, building on the foundational work of Ho and Stoll, and later Avellaneda and Stoikov, the academic literature on market making has expanded to address trade size distributions, client tiering, complex price dynamics, alpha signals and the internalisation versus externalisation dilemma in markets with dealer-to-client and interdealer-broker segments. In this paper, we tackle two critical dimensions: adverse selection, arising from the presence of informed traders, and price reading, whereby the market maker's own quotes reveal the direction of their inventory. These risks are well known to practitioners, who routinely face informed flows and algorithms capable of extracting signals from quoting behaviour. Yet they have received limited attention in the quantitative finance literature, beyond stylised toy models with limited actionability. Extending the existing literature, we propose a tractable framework that enables market makers to adjust their quotes with greater awareness of informational risk.

Figures

Figures reproduced from arXiv: 2508.20225 by the authors.

**Figure 1.** Optimal bid quote adjustment, ε −1 bd n,k,b∗ (q) − bδ n,k,b∗ (q) , with the second tier subject to price reading (we illustrate the first and the fourth sizes) – SVS = 25%. 60 40 20 0 20 40 60 Inventory (M) 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 Bid quote adjustment 1, k = 1 M 1, k = 10 M 2, k = 1 M 2, k = 10 M [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗

**Figure 2.** Optimal bid quote adjustment, ε −1 bd n,k,b∗ (q) − bδ n,k,b∗ (q) , with the second tier subject to price reading (we illustrate the first and the fourth sizes) – SVS = 75%. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Average values of the simulated market maker’s PnL as a function of safe volume share for the different strategies. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Values of the standard deviation for the simulated market maker’s PnL as a function of safe volume share for the [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: Risk-adjusted performance measure associated with the simulated market maker’s PnL as a function of safe volume [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: Optimal bid quote adjustment, ε −1 bd n,k,b∗ (q) − bδ n,k,b∗ (q) , with the second tier subject to adverse selection (we illustrate the first and the fourth sizes) – SVS = 50% [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: Average values of the simulated market maker’s PnL as a function of safe volume share, comparing the “Optimal” [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

**Figure 8.** Figure 8: Values of the standard deviation for the simulated market maker’s PnL as a function of safe volume share, comparing [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

**Figure 9.** Figure 9: Risk-adjusted performance measure associated with for the simulated market maker’s PnL as a function of safe volume [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

Discussion (0). Continue with ORCID to comment.

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