{"id":"2a1b24bb-62ea-4db7-9827-3511d2512c97","arxiv_id":"2508.20225","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"An optimal market-making model with informed order flow and skew-sniffing that yields first-order quote corrections per client tier.","lead":"Dealers who post bid and ask prices can be hurt by clients who trade on superior information and by 'skew sniffers' who guess the dealer's position from the quote ladder. This paper derives simple quote-adjustment formulas for both risks and tests them in simulations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"First-order expansion is run at ε=1 where corrections are as large as baseline quotes (Figs. 1–2, footnote 9), so Eqs. (12)–(13) lack error control and the simulated 'Optimal' PnL gains are not evidence of optimality.","rationale":"The reader's weakest assumption targets exactly the right soft spot: the perturbation parameter ε is introduced as small but is set to 1 in the simulations, with no uniform error control. My reading of Section 3.3.3 and Section 5 confirms this is load-bearing: Eqs. (12)–(13) are derived as first-order in ε, and the simulations evaluate them at ε=1, where the plotted adjustments are the same order of magnitude as the baseline quotes. The paper itself flags the regime in footnote 9. Without a numerical or analytic check against the true HJB solution, the simulated 'Optimal' strategy cannot be certified as approximately optimal. This does not undermine the formal first-order derivation, which appears internally consistent, nor the paper's honesty about limitations; it just leaves the central practical claim conditional. A full HJB policy-iteration comparison is the natural, feasible test.","tokens_in":17829,"tokens_out":12251,"duration_ms":143062,"concrete_test":"Solve the full HJB equation (6) numerically for the Section 5.2/5.3 calibration with ε=1 and the floor δ≥0: policy iteration on q∈[-100,100]M with Δq=1M (check truncation at ±200M) using functional forms (14). Compare the resulting optimal quotes to the approximate formulas (12)–(13) pointwise in q and for each tier/size. Then run the paper's 10^5-path Monte Carlo for the true optimal policy and the approximate 'Optimal' policy and compare mean PnL, std, and ratio. If the approximate strategy's risk-adjusted PnL is within, say, 5% of the true optimal, the concern is resolved; if it is materially worse, Eqs. (12)–(13) do not support the claimed optimal quoting. A secondary check: repeat with ε=0.1 and ε=0.5 to estimate the radius of validity of the expansion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central output is Eqs. (12)–(13), first-order expansions of optimal quotes in ε around the no-information baseline. For these to describe optimal quoting, the o(ε) remainder must be uniformly small over the inventory/quote range used. The paper provides no remainder bound. Worse, the numerical section sets ε=1 and uses the same expansion for the closed-form 'Optimal' strategy, and footnote 9 admits a 0-floor is needed because 'the applicability of the first-order approximations is questionable' in extreme situations. The magnitudes confirm this: with κ=3bp^-1, baseline half-spread is ~1/κ=0.33bp, while Figures 1–2 show quote adjustments of order 1bp, i.e., corrections comparable to (or larger than) the baseline quotes. Thus the expansion is not a small perturbation at the simulated operating point. The simulated PnL improvements of 'Optimal' over No Action/No Skew are therefore not evidence that (12)–(13) approximate the true optimal controls; they only show that one particular heuristic outperforms two other heuristics inside the same model. The model-based PnL comparison also cannot distinguish 'closer to optimal' from 'different but lucky', so the central claim of implementable optimal quoting is under-supported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":18269,"tokens_out":5985,"duration_ms":67807,"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":[{"comment":"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.","section":"§5.2–5.3, footnote 9, Eqs. (12)–(13)"},{"comment":"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).","section":"§3.3.2–§3.3.3, Eq. (7), Eq. (6)"},{"comment":"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.","section":"§5.2–§5.3, Figs. 3–9"}],"minor_comments":[{"comment":"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.","section":"§5.3, Figs. 7–9"},{"comment":"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.","section":"Eq. (7) and §3.3.2"},{"comment":"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.","section":"§4.2, Eqs. (9)–(10)"},{"comment":"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.","section":"§5.1"}],"recommendation":"major_revision","confidential_remarks":"This is a well-written extension of the authors' earlier quadratic-Hamiltonian method [6], and the closed-form formulas are genuinely useful. The central issue is the numerical section's use of ε=1 without error control, which undermines the 'optimal' wording. I would ask the authors to either provide a rigorous remainder bound (unlikely to be easy) or, more realistically, to reframe the contribution as a first-order approximation/heuristic and strengthen the numerical evidence with a small-ε convergence test or a comparison against a full HJB solver. The citation to an unpublished preprint [10] is a minor point and not a concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nShould 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.\n\nSo: 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.","headline":"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.","tokens_in":18681,"tokens_out":1295,"would_cite":true,"duration_ms":15705,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91G80","93E20"],"pacs":[],"model":"deepseek-v4-flash","headline":"A first-order formula gives market makers explicit quote corrections for informed flow and skew sniffers, split into global and tier-specific parts.","keywords":["market making","adverse selection","price reading","informational risk","stochastic optimal control","perturbation analysis","quote skew","client tiering"],"falsifier":"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.","tokens_in":17701,"feed_emoji":"💹","tokens_out":6078,"duration_ms":63097,"temperature":0.7,"pith_summary":"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.","feed_headline":"Dealers get explicit quote formulas for toxic flow and skew sniffers","feed_subtitle":"A first-order expansion decomposes quote changes into a global risk term and a per-tier de-skewing term, improving risk-adjusted PnL.","key_machinery":"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.","core_discovery":"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,","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the baseline intensity-based market making model with inventory penalty that the paper extends.","marker":"[1]"},{"why":"Provides the foundational optimal dealer pricing framework under transaction and return uncertainty.","marker":"[15]"},{"why":"Establishes the market making objective and Hamiltonian approach used for the no-informational-risk baseline.","marker":"[13]"},{"why":"Introduces the Cartea-Jaimungal style risk-adjusted expected PnL objective and related stochastic control setting.","marker":"[8]"},{"why":"Supplies the quadratic Hamiltonian approximation method used to obtain closed-form corrections in Section 4.","marker":"[6]"},{"why":"Motivates the adverse selection channel through the classic theory of bid-ask spreads with informed traders.","marker":"[11]"},{"why":"Documented practitioner evidence for 'skew sniffers', motivating the price reading effect.","marker":"[12]"},{"why":"Provides economic evidence on information leakage in competition, supporting the practical relevance of price reading.","marker":"[2]"}],"fun_headline_variants":["Explicit optimal quotes for informed flow and quote-sniffing","New formulas break down quote risk into global and per-tier terms","First-order fix for adverse selection and price reading in market making","Tractable model gives closed-form quote adjustments for toxic order flow","Market makers get decomposable corrections for informed traders and quote readers"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Explicit optimal quotes for informed flow and quote-sniffing","New formulas break down quote risk into global and per-tier terms","First-order fix for adverse selection and price reading in market making","Tractable model gives closed-form quote adjustments for toxic order flow","Market makers get decomposable corrections for informed traders and quote readers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001203,"raw_usage":{"total_tokens":4777,"prompt_tokens":713,"completion_tokens":4064,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":457,"completion_tokens_details":{"reasoning_tokens":3977}},"tokens_in":457,"tokens_out":4064,"duration_ms":30122,"temperature":1.0,"reasoning_tokens":3977,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T15:16:35.383912+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"CRC Press, 2016","cited_arxiv_id":null,"evidence_quote":"Establishes the market making objective and Hamiltonian approach used for the no-informational-risk baseline."},{"cited_title":"Cambridge University Press, 2015","cited_arxiv_id":null,"evidence_quote":"Introduces the Cartea-Jaimungal style risk-adjusted expected PnL objective and related stochastic control setting."},{"cited_title":"Closed-form approximations in multi-asset market making.Applied Mathematical Finance, 28(2):101–142, 2021","cited_arxiv_id":null,"evidence_quote":"Supplies the quadratic Hamiltonian approximation method used to obtain closed-form corrections in Section 4."},{"cited_title":"Glosten and Paul R","cited_arxiv_id":null,"evidence_quote":"Motivates the adverse selection channel through the classic theory of bid-ask spreads with informed traders."},{"cited_title":"BNPP ups efforts to weed out skew sniffers.FX Markets, 20 Nov 2024","cited_arxiv_id":null,"evidence_quote":"Documented practitioner evidence for 'skew sniffers', motivating the price reading effect."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides economic evidence on information leakage in competition, supporting the practical relevance of price reading."}],"review_version":1}