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Avellaneda-Stoikov and Cartea-Jaimungal as One Framework: A Forced Uniqueness Theorem for Inventory Market Making

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Pith's one-line read Axioms on dynamic preferences force the Avellaneda-Stoikov and Cartea-Jaimungal market-making frameworks to be two views of the same entropic object with linked parameters.

desk verdict The paper derives that five axioms force the Avellaneda-Stoikov objective to be the unique entropic form and Cartea-Jaimungal to be its inventory Taylor expansion with φ tied to γ, but the terminal coefficient link needs an extra regularity condition whose scope is unclear from the abstract. read the letter →

arxiv 2606.01477 v3 pith:UCS23Y4V submitted 2026-05-31 q-fin.MF q-fin.RMq-fin.TR

Frank M. V. Feys This is my paper · ORCID
classification q-fin.MFq-fin.RMq-fin.TR
keywords inventorymarketmakingentropiccertaintyequivalentdynamicconsistencypreferenceaxiomsAvellaneda-StoikovmodelCartea-Jaimungalliquidationpenaltyparameters
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

The paper establishes that cash-additivity, normalization, concavity, strong dynamic consistency, and law-invariance on a market maker's preference functional force it to take the specific form of an entropic certainty equivalent applied to liquidation-adjusted terminal wealth, parametrized by one scalar γ. This axiom set makes the Avellaneda-Stoikov model the unique representative inside the class. The Cartea-Jaimungal model then appears as the second-order Taylor expansion of that same functional in inventory size, which pins the running penalty coefficient at φ = γ σ² / 2 and, under a regularity condition, the terminal coefficient at α = ½ L''(0). A reader would care because the two frameworks are thereby shown to be different approximations of one underlying preference rather than independent modeling choices that can be calibrated separately.

What carries the argument

The entropic certainty-equivalent on liquidation-adjusted terminal wealth, parametrized by a single positive scalar γ, which encodes the entire preference structure.

What would settle it

Independent calibration of both frameworks on identical market data, followed by a direct check whether the fitted values satisfy φ = γ σ² / 2 within statistical error; systematic violation on multiple assets or periods would falsify the forced uniqueness.

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Extended reading notes

Core claim

Under the five listed axioms the preference functional is forced to be the entropic certainty-equivalent on liquidation-adjusted terminal wealth parametrized by a single positive scalar γ; the Avellaneda-Stoikov framework is therefore the unique model in this class, while the Cartea-Jaimungal framework is its second-order Taylor expansion in inventory magnitude with the running coefficient forced to φ = γ σ² / 2 and the terminal coefficient forced to α = ½ L''(0) under the stated regularity condition on the liquidation cost.

Load-bearing premise

The mild regularity condition on the liquidation cost function that is used to force the terminal coefficient α to equal ½ L''(0).

Editorial extensions

If this is right

  • The two frameworks cannot be treated as competing alternatives whose choice is driven only by tractability; they are different manifestations of one preference object.
  • The relation γ = 2 φ / σ² supplies an immediate consistency cross-check on any pair of independently calibrated desk parameters.
  • Higher-order expansions of the same entropic functional would generate further market-making approximations whose coefficients are likewise determined by the single scalar γ.
  • Any model that preserves the five axioms must reproduce the same entropic form and therefore the same parameter linkage.

Reading between the lines

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

  • If the relation is observed to hold in practice, it would suggest that real market-maker risk preferences are close to entropic on liquidation-adjusted wealth.
  • The unification opens the possibility of importing approximation techniques or numerical methods from one literature directly into the other without re-calibrating free parameters.
  • Extensions to multi-asset or stochastic-volatility settings could be tested for consistency by checking whether the same γ continues to link the running and terminal penalties across assets.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 0 minor

Summary. The paper claims that five axioms on the market maker's dynamic preference functional (cash-additivity, normalization, concavity, strong dynamic consistency, and law-invariance) force the functional to be the entropic certainty-equivalent on liquidation-adjusted terminal wealth, parametrized by a single positive scalar γ. The Avellaneda-Stoikov framework is the unique representative of this axiom class. The Cartea-Jaimungal framework is its second-order Taylor expansion in inventory magnitude, with the running coefficient forced to φ = γσ²/2 and (under a mild regularity condition on the liquidation cost function) the terminal coefficient forced to α = ½L''(0). The forced relation is invertible, γ = 2φ/σ², providing a consistency cross-check on independently calibrated parameters.

Significance. If the result holds, the paper unifies two standard inventory market-making frameworks by deriving them from the same axiomatic preference structure, showing they are manifestations of a single object rather than competing alternatives. This supplies a theoretical basis for relating their free parameters and a practical cross-check on desk calibrations. The axiomatic derivation yielding a parameter-free uniqueness result and the explicit invertible relation between γ and φ are strengths that would be valuable to the mathematical finance literature on market making.

major comments (2)
  1. [Abstract (Taylor expansion paragraph)] Abstract, paragraph on the Taylor expansion: The relation α = ½ L''(0) is qualified by a 'mild regularity condition on the liquidation cost function.' The manuscript must explicitly define this condition (e.g., twice differentiability of L at zero and isolation of the second-derivative term) and state the consequences if it fails, because the uniqueness theorem itself does not invoke the condition; without it the terminal penalty in the Cartea-Jaimungal approximation is no longer pinned by the same γ, so the 'single underlying object' claim holds only for the running penalty φ.
  2. [Uniqueness theorem derivation] Derivation of the entropic form (section containing the uniqueness theorem): The abstract asserts that the five axioms force the entropic certainty-equivalent. The full proof steps from strong dynamic consistency and law-invariance to this specific functional form must be checked to confirm that the subsequent Taylor-expansion step does not introduce post-hoc restrictions that affect the central claim relating the two frameworks.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points on clarity and the precise scope of the uniqueness result versus its approximation. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract (Taylor expansion paragraph)] Abstract, paragraph on the Taylor expansion: The relation α = ½ L''(0) is qualified by a 'mild regularity condition on the liquidation cost function.' The manuscript must explicitly define this condition (e.g., twice differentiability of L at zero and isolation of the second-derivative term) and state the consequences if it fails, because the uniqueness theorem itself does not invoke the condition; without it the terminal penalty in the Cartea-Jaimungal approximation is no longer pinned by the same γ, so the 'single underlying object' claim holds only for the running penalty φ.

    Authors: We agree that the mild regularity condition requires explicit definition and that its failure affects only the terminal coefficient. We will revise the abstract and the relevant section on the Taylor expansion to state the condition as twice continuous differentiability of the liquidation cost L at zero (with the second-derivative term isolated after the first-order term vanishes by normalization). If the condition fails, the terminal penalty α in the Cartea-Jaimungal approximation is no longer forced to equal ½L''(0) by the same γ, while the running-penalty relation φ = γσ²/2 continues to hold unconditionally from the second-order expansion of the entropic functional. The uniqueness theorem itself remains unaffected, as it concerns only the entropic form; the 'single underlying object' claim will be qualified accordingly in the revised text. revision: yes

  2. Referee: [Uniqueness theorem derivation] Derivation of the entropic form (section containing the uniqueness theorem): The abstract asserts that the five axioms force the entropic certainty-equivalent. The full proof steps from strong dynamic consistency and law-invariance to this specific functional form must be checked to confirm that the subsequent Taylor-expansion step does not introduce post-hoc restrictions that affect the central claim relating the two frameworks.

    Authors: The uniqueness theorem is proved in Section 3. The argument proceeds by first invoking cash-additivity, normalization and concavity to obtain a concave monetary utility functional, then applying strong dynamic consistency to obtain a recursive representation, and finally using law-invariance to reduce the problem to a static entropic form on the terminal liquidation-adjusted wealth; the resulting functional is necessarily the entropic certainty equivalent parametrized by a single γ > 0. The Taylor-expansion step that produces the Cartea-Jaimungal running and terminal penalties is applied only after the uniqueness result has been established and does not feed back into the axiomatic derivation. Consequently, no post-hoc restrictions are introduced. We are prepared to expand the proof steps in an appendix if the editor requests further detail, but the existing derivation already separates the axiomatic uniqueness from the subsequent approximation. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation from external axioms to entropic form and Taylor relations is self-contained with no reduction by construction.

full rationale

The paper states a list of axioms (cash-additivity, normalization, concavity, strong dynamic consistency, law-invariance) and claims they force the preference functional to be the entropic certainty-equivalent parametrized by γ; the AS framework is declared the unique representative and CJ its second-order expansion with φ = γσ²/2 and α = ½L''(0) under a regularity condition. No step reduces to a self-definition, a fitted input relabeled as prediction, or a load-bearing self-citation whose content is itself unverified. The uniqueness theorem and expansion are presented as derived within the paper rather than imported from overlapping prior work by the same author. The regularity condition is an explicit additional assumption, not a hidden tautology. The central claim therefore retains independent content relative to its inputs.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the five named axioms from dynamic preference theory plus one mild regularity condition on the liquidation cost; γ is the sole free parameter once the axioms are accepted.

free parameters (1)
  • γ
    Single positive scalar that parametrizes the entropic certainty-equivalent once the axioms are imposed.
assumptions (5)
  • domain assumption cash-additivity
    Invoked to force the preference to depend only on terminal wealth after liquidation.
  • domain assumption normalization
    Standard normalization axiom used to pin the functional form.
  • domain assumption concavity
    Used to obtain the entropic representation.
  • domain assumption strong dynamic consistency
    Central axiom that eliminates time-inconsistent alternatives.
  • domain assumption law-invariance
    Ensures the functional depends only on the distribution of terminal wealth.

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Cite this review

Pith. "Pith review of Avellaneda-Stoikov and Cartea-Jaimungal as One Framework: A Forced Uniqueness Theorem for Inventory Market Making." pith.science (2026). https://pith.science/paper/UCS23Y4V

@misc{pith2026260601477,
  author       = {Pith},
  title        = {Pith review of: Avellaneda-Stoikov and Cartea-Jaimungal as One Framework: A Forced Uniqueness Theorem for Inventory Market Making},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UCS23Y4V}},
  note         = {Machine review of arXiv:2606.01477}
}
abstract

In inventory market making, the running-penalty coefficient $\phi$ of the Cartea-Jaimungal framework and the risk-aversion parameter $\gamma$ of the Avellaneda-Stoikov framework are typically treated as independent free parameters, calibrated separately. We show that they are in fact not independent. A small set of axioms on the market maker's dynamic preference functional, namely cash-additivity, normalization, concavity, strong dynamic consistency, and law-invariance, forces the preference functional to be the entropic certainty-equivalent on liquidation-adjusted terminal wealth, parametrized by a single positive scalar $\gamma$. The Avellaneda-Stoikov framework is the unique representative of this axiom class. The Cartea-Jaimungal framework is its second-order Taylor expansion in inventory magnitude, with the running coefficient forced to $\phi = \gamma\sigma^2/2$ and (under a mild regularity condition on the liquidation cost) the terminal coefficient forced to $\alpha = \frac{1}{2}L''(0)$. The two frameworks, typically presented as competing alternatives with the choice between them driven by tractability, are different manifestations of a single underlying object. The forced relation is invertible, $\gamma = 2\phi/\sigma^2$, giving a consistency cross-check on independently calibrated desk parameters.

Figures

Figures reproduced from arXiv: 2606.01477 by the authors.

Figure 1
Figure 1. The forced relation ϕ = γσ2/2 in two views. (a) The constraint surface in parameter space; off-surface calibrations are over-parametrized. (b) The stochastic-volatility version, ϕt = γσ2 t /2, tracks realized variance pointwise in time, while a wall-clock constant systematically misprices inventory. 42 [PITH_FULL_IMAGE:figures/full_fig_p042_1.png] view at source ↗

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Cited by 1 Pith paper

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