Pith. sign in

REVIEW 4 major objections 6 minor 62 references

Learning Market Making with Closing Auctions

T0 review · 4 major / 6 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A market-making agent that explicitly tracks a projected closing-auction clearing price while posting quotes on the continuous order book can earn higher mean returns than standard benchmarks.

desk verdict Auction-aware MDP formulation is novel and the clearing-price theorem is fine, but the reported outperformance is not credible because the episode 'return' includes non-cash fictive auction rewards. read the letter →

arxiv 2601.17247 v2 pith:M7G5CSSR submitted 2026-01-24 q-fin.TR math.OC

Julius Graf , Thibaut Mastrolia This is my paper · ORCID
classification q-fin.TRmath.OC MSC 91G8068T0590C40
keywords optimalmarketmakingclosingauctionreinforcementlearningMarkovDecisionProcessDeepQ-Learningclearingpricelimitorderbookregretanalysis
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 claims that a market maker can improve expected profit by explicitly anticipating the closing auction during continuous limit-order-book trading, rather than treating end-of-day liquidation as a static penalty. It proposes a reinforcement-learning framework where a neural-fitted Q-learning agent learns to quote in the continuous phase and to submit, cancel, and price orders in the closing auction, using a continuously updated estimate of the eventual clearing price. The authors build a generative market simulator and test the learned policy against two classical benchmarks, using both a rough-Heston price model and historical S&P 500 price paths. In both settings, they report that the learned policy outperforms the benchmarks on mean returns, with the auction phase contributing substantially to the profit.

What carries the argument

The central object is the projected (hypothetical) clearing price, H^cl_t: during the continuous phase it is computed by treating outstanding limit orders as a fictitious auction (Algorithm 1), and during the auction phase it is the solution of the estimated clearing-price equation (2). This single quantity links the two trading phases: it drives the reward (executions priced below H^cl_t are penalized), determines the agent's auction quotes (linear supply curve with slope K and reference price S), and enters the final PnL. The paper also proves Theorem 2.1, giving sufficient conditions (continuity, monotonicity, and a contraction condition) for existence and uniqueness of the estimated clea

What would settle it

Run the same algorithm with the agent receiving only publicly available auction imbalance information (instead of the full exogenous order book) for both training and evaluation on the same simulator and S&P 500 paths, and check whether the mean-return advantage over the two benchmarks persists; if it disappears, the full-information assumption is the source of the reported gain.

Watch

Extended reading notes

Core claim

The paper's central claim is that the closing auction should be a first-class object in market-making decisions, not an afterthought for inventory liquidation. To that end, it introduces a projected clearing price H^cl_t, which the agent updates throughout the continuous phase by treating standing limit orders as a hypothetical auction, and which it refines during the auction phase by solving the estimated clearing-price equation. The policy is learned with neural-fitted Q-learning, with separate networks for the two phases, and the reward design explicitly penalizes executions below the projected clearing price and wrong-side dealing in the auction. On evaluation over 100 fresh episodes, th

Load-bearing premise

The agent is assumed to have full information on all exogenous auction limit and market orders (Section 2.1.2), so the projected clearing price is computed from exactly the order flow that determines the final clearing price in simulation; real closing auctions do not afford such transparency, and without it the claimed outperformance is not established.

Editorial extensions

If this is right

  • Market makers should condition their continuous-phase quotes on a running estimate of the auction clearing price instead of only on the book's mid price.
  • A large share of the learned policy's return comes from the auction phase (mean auction reward turns positive after training), so the auction is a profit source, not just a liquidation venue.
  • The same neural-fitted Q-learning setup, with unchanged hyperparameters, works on both generative rough-Heston data and historical S&P 500 paths, suggesting the approach is not tied to a single price process.
  • Theorem 2.1 supplies general conditions under which a unique estimated clearing price exists, extending auction-clearing computations beyond the linear supply/demand case used in the numerical experiments.

Reading between the lines

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

  • The full-information assumption (the agent sees every exogenous auction order) means the projected clearing price in the simulator is nearly an oracle; on real markets, where only indicative imbalance is visible, the reported margin may narrow.
  • Part of the gap versus the classical benchmark likely reflects that the benchmark is derived for a Bachelier mid-price while the test data are rough-Heston or S&P 500; a sharper test would benchmark against the true optimal control under rough volatility.
  • The framework suggests a general recipe: for any session that ends in a discrete liquidity event (periodic auctions, call markets, workup mechanisms), make the projected clearing price a state variable and let RL optimize across the two phases.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper studies a market-making execution problem on a session consisting of a continuous limit-order-book phase followed by a closing auction. It introduces a mathematical framework for auction clearing, a projected clearing price H^cl, and a Markov Decision Process whose rewards cover both trading phases. The proposed method is Neural Fitted Q-iteration (NFQ), and the authors compare it against an Avellaneda–Stoikov benchmark and a TWAP benchmark. The central claim is that the trained NFQ policy outperforms both benchmarks on mean returns, in simulations driven by a rough-Heston mid-price model and by historical S&P 500 mid prices. The paper also proves existence and uniqueness of a projected clearing price under general supply/demand curves (Theorem 2.1).

Significance. The topic is timely and practically relevant: closing auctions concentrate a large fraction of daily volume, and most existing market-making models ignore them. The paper is transparent about its generative model, hyperparameters, and algorithm details, which is a strength. Theorem 2.1 is coherent, though the contraction-mapping argument is standard. The main empirical result, however, is not yet convincing: the reported returns mix realized PnL with non-cash fictive auction rewards; the agent is given full information on the order flow that determines the clearing price; and the evaluation lacks out-of-sample testing and statistical significance. Because these issues bear directly on the headline claim, the paper needs substantial revision before the main contribution can be assessed.

major comments (4)
  1. [§3, §6.3.3, Table 3; §6.4, Table 4] The 'Mean Return' used to claim outperformance is the sum of MDP rewards, which include the per-step auction reward r_t = K^a_t H^cl_t(H^cl_t − S^a_t) + f_a(...) − d‖c_t‖_1 for τop ≤ t < τcl (§3, reward case 2). Remark 6.1 acknowledges that the agent can obtain 'fictive rewards' in the auction. Since the agent has full information on exogenous auction orders (§2.1.2), H^cl_t is the projected clearing price computed from exactly the order flow that determines the final clearing price (Eq. (2)), and the agent's own orders feed back into H^cl_t. Tables 3 and 4 do not separate these interim fictive rewards from the terminal settlement PnL at τcl. The outperformance may therefore be an artifact of reward shaping rather than a genuine increase in market-making profit, even inside the simulator. Please report the realized terminal PnL (cash settlement minus inventory cost) separately and show t
  2. [§2.1.2, §2.1.4] The agent is assumed to 'have access to full information' on the activities of all exogenous auction participants. In real closing auctions, participants do not observe the complete order flow; the projected clearing price H^cl_t is not computable from public information. The learned policy's advantage appears to rely on this unrealistic assumption, because H^cl_t is used both as a state variable and as the reward-shaping signal. An ablation where the agent observes only public data (e.g., indicative uncrossing price, imbalance, past cancellations) is needed to establish that the approach does not depend on the full-information assumption.
  3. [§6.3, §6.4] The policy is trained and evaluated on the same generative model; for the historical-data experiment, the same realized mid-price path is reused for all training episodes and evaluation episodes. No out-of-sample test across different days or market conditions is provided. In addition, Tables 3 and 4 report no standard errors or p-values; with the reported standard deviations (Final NFQ std 16,552 versus mean 12,169 in Table 3), the mean differences may be within sampling noise. Please provide bootstrap confidence intervals for the mean-return differences and an out-of-sample, or at least train/test split, evaluation.
  4. [§5.1, §6.3.1] The AS benchmark is derived under an arithmetic Brownian motion mid-price with constant transaction size, and is then applied in a discrete-time, rough-Heston simulation. Section 6.3.1 states: 'We are in fact expecting our model to beat the benchmark (since the benchmark is only optimal for a Bachelier process)'. Consequently, outperforming AS in the synthetic setting does not establish superiority over an optimal auction-aware market maker; it demonstrates superiority over a deliberately misspecified baseline. This limitation should be acknowledged in the abstract and conclusions.
minor comments (6)
  1. [§1.3] Typo: 'closing auciton' should be 'closing auction'.
  2. [§4.1] The symbol T is used both for the time grid (§2.1) and for the total number of time steps T = (m+2)E in Definition 4.1. Please use different notation to avoid ambiguity.
  3. [Table 2] M1=10 and M2=-10 are labeled 'upper bound' and 'lower bound', but the sampler S^i_t ∼ S_mid + αU({M1,...,M2}) implies the support is {10,...,-10} if taken literally. Please clarify the intended support (e.g., M1=-10, M2=10).
  4. [Algorithm 2, line 5] The notation V^{ζ,j}_t = ρ^{-1} V^{ζ,j+1}_t is ill-defined at j=L and appears inverted; presumably V^{ζ,j+1}_t = ρ V^{ζ,j}_t for j=1,...,L-1.
  5. [Table 3] Several rows contain only two numerical entries, making it difficult to compare all four strategies. Please align the table layout so that each row has one entry per strategy column.
  6. [§4] The paper promises 'regret analysis' in the introduction (Section 1.3), but the body only defines pseudo-regret and plots it numerically; no theoretical regret bound for NFQ is provided. Consider rewording the section title or adding a formal statement.

Circularity Check

1 steps flagged · score 6.0 of 10

Numerical 'outperformance' is measured on the same reward objective the NFQ policy is trained to maximize, with fictive auction rewards driven by full-information projected clearing prices.

  1. fitted input called prediction [Section 3 (reward definition), Remark 6.1, Section 6.3.3 Table 3, Section 6.4 Table 4]
    "The agent receives a fictive reward K^a_t H^cl_t(H^cl_t - S^a_t), where H^cl_t is the anticipated clearing price (were the auction to close at time t). ... We still expect the model to outperform benchmarks on average but with higher variance, given the low penalty parameters and the agent's ability to obtain fictive rewards in the closing auction."

    The paper reports 'Mean Return' in Tables 3 and 4 as the performance metric, but this 'return' is the sum of MDP rewards defined in Section 3, which is exactly the objective the NFQ policy is trained to maximize. Moreover, the auction-phase reward contains a 'fictive' term K^a_t H^cl_t(H^cl_t - S^a_t), where H^cl_t is not an independent forecast but the solution of Equation (2) computed from full information on all exogenous orders plus the agent's own orders. The policy can therefore inflate the reported metric by choosing K^a and S^a to manipulate H^cl_t, and the claimed outperformance is partly an artifact of optimizing the reported metric, not a prediction of actual PnL. The paper's own Remark 6.1 concedes the role of 'fictive rewards.'

full rationale

The theoretical content (Theorem 2.1, Corollary 2.1, and the benchmark derivations in Section 5) is self-contained and not circular: the clearing-price existence claim is proved via a contraction argument, and the AS/TWAP benchmarks are standard external models. However, the central numerical claim—that the NFQ policy 'outperforms the two benchmarks on mean returns'—is circular in a narrower sense. The reported 'mean returns' are the very MDP rewards the policy was trained to maximize, and the auction component of those rewards is explicitly 'fictive' and depends on a projected clearing price computed from full information on the order flow that determines it. Thus the numerical superiority is, to a significant degree, a check that the optimizer improved its training objective rather than an independent test of profitability. Self-citations to [19], [41], [42], [52] are present, but they support modeling choices (e.g., linear supply curves, auction-price calibration) rather than the uniqueness or validity of the core result, so I do not treat them as independently circular. Because one central 'prediction' reduces by construction to the fitted reward objective, the circularity score is 6.

Assumptions & free parameters 19 free parameters · 10 assumptions · 1 invented entities

The central numerical claim rests on a large set of hand-set generative parameters, strong information assumptions, and an ad hoc projected-clearing-price construct. The only externally grounded component is the standard Q-learning convergence theorem and the prior calibration of the rough Heston model, but those do not validate the simulated auction mechanism.

free parameters (19)
  • λ0 (market taker arrival intensity) = 1
    Poisson intensity for buy/sell market orders in the continuous phase; hand-set in Table 2.
  • v_m, γ_m (Pareto volume parameters) = 2, 2.5
    Shape the heavy-tailed distribution of market order and LOB volumes; hand-set.
  • V∞, βa, βb (Beta LOB volume parameters) = 15, 2, 5
    Generate first-level LOB volumes in the emulator; hand-set.
  • ρ (LOB geometric decay) = 0.5
    Controls depth decay of the limit order book; hand-set.
  • V (maximum market order volume) = 30
    Caps sampled order volumes; hand-set.
  • U1, U2 (auction supply slope bounds) = 0.1, 2.0
    Bounds for exogenous auction-maker supply slopes; hand-set.
  • M1, M2 (auction reference price bounds) = 10, -10
    Bounds for exogenous auction reference prices around the mid price; hand-set.
  • p1, p2, p3, p4 (auction event probabilities) = 0.3, 0.2, 0.3, 0.1
    Bernoulli probabilities for auction market maker/taker arrival and cancellation; hand-set.
  • λ (inventory penalty) = 0.5
    Terminal inventory penalty in the final reward; hand-set.
  • q (wrong-side dealing penalty) = 1
    Penalty factor for auction orders that deal on the wrong side; hand-set.
  • k⋆ (tolerance) = 1000
    Tolerance in the continuous-phase reward comparing execution price with projected clearing price; hand-set.
  • d (cancellation cost) = 0.1
    Per-unit cost of canceling auction orders; hand-set.
  • α (tick size) = 0.01
    Tick size of the asset; hand-set.
  • β, K (auction slope grid) = 3.33, 10
    Grid step and upper bound for the agent's auction supply slope; hand-set.
  • τ, H0 (Algorithm 1 smoothing and initial clearing price) = 0.95, 100
    Smoothing parameter and initial value for the projected hypothetical clearing price; hand-set.
  • NFQ hyperparameters = η=3e-4, M=3, N=50k, N_min=5k, χ=0.99, E=2000, B=128
    Learning rate, epochs, buffer sizes, discount, episodes, and batch size; chosen by the authors and not justified by sensitivity analysis.
  • σ (AS benchmark volatility) = estimated per asset (e.g., 5.77e-4 for CAT)
    Fitted from the same simulated or historical mid-price paths used for evaluation.
  • K (AS price-impact constant) = from least-squares regression on 5,000 simulated LOB samples
    Fitted to the generative model, so the AS benchmark is not independent of the simulated environment.
  • Rough Heston parameters = H=0.1, ρ=-0.7, V0=0.02, θ=0.04, λ=0.3, ν=0.3
    Taken from a prior calibration [1] and scaled to the trading period; affects all synthetic mid-price paths.
assumptions (10)
  • domain assumption Assumption 1: all market orders are always executed during the LOB phase.
    Stated in Section 2.1.1; used to justify the execution model and the non-empty LOB.
  • domain assumption Assumption 2: the agent always has execution priority at a fixed depth of the CLOB.
    Stated in Section 2.1.1; makes the agent a high-frequency trader with time priority.
  • domain assumption Assumption 3: the agent can only act at discrete times during the auction.
    Stated in Section 2.1.2; defines the MDP time grid.
  • domain assumption Assumption 4: the agent's auction supply curve is linear.
    Stated in Section 2.1.2; used in Proposition 2.1 and the reward design.
  • domain assumption Assumption 5: at least one new market taker arrives on each side each discrete period.
    Stated in Section 3; ensures every action has an effect on the next state.
  • domain assumption The agent has full information on all exogenous auction limit and market orders.
    Assumed in Section 2.1.2; powers the projected clearing price computation and is not realistic for real auctions.
  • domain assumption Exogenous auction supply functions g_i are continuous, increasing, with limits ±∞, and satisfy a uniform Lipschitz condition.
    Assumptions in Theorem 2.1 for existence and uniqueness of the clearing price.
  • ad hoc to paper Generative model distributional choices: Pareto, Beta, Bernoulli, and uniform sampling.
    Section 6.1; none of these distributions is derived from real closing-auction data.
  • standard math Q-learning converges when rewards are bounded and learning rates satisfy the Robbins-Monro conditions.
    Invoked from [62]; used to justify the NFQ approach, though no convergence proof is given for the neural version.
  • ad hoc to paper NFQ with a neural network converges to an optimal policy.
    Assumed without proof; the paper's loss plots show non-stabilized auction loss after episode 1,000.
invented entities (1)
  • Hcl — projected hypothetical clearing price
    purpose: State variable used during continuous trading to anticipate the closing auction and to shape the reward function.
    Computed by Algorithm 1 from simulated standing LOB volumes; no real-market validation, and it directly influences the agent's training signal.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Learning Market Making with Closing Auctions." pith.science (2026). https://pith.science/paper/M7G5CSSR

@misc{pith2026260117247,
  author       = {Pith},
  title        = {Pith review of: Learning Market Making with Closing Auctions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7G5CSSR}},
  note         = {Machine review of arXiv:2601.17247}
}
read the original abstract

In this work, we investigate a market making execution problem on a trading session in which a continuous phase on a limit order book is followed by a closing auction. Whereas standard optimal market making models typically rely on terminal inventory penalties to manage end-of-day risk, ignoring the significant liquidity events available in closing auctions, we propose a deep reinforcement learning framework, consisting of a Deep Q-Network and its continuous-control actor-critic extensions (DDPG, TD3 and SAC), that explicitly incorporates this mechanism. We introduce a market making framework designed to explicitly anticipate the closing auction, continuously refining the projected clearing price as the trading session evolves. We develop a generative stochastic market model to simulate the trading session and to emulate the market. Our theoretical model and these deep reinforcement learning methods are applied on the generator in two settings: (1) when the mid price follows a rough Heston model with generative data from this stochastic model; and (2) when the mid price corresponds to historical data of assets from the S&P 500 index and the performance of our algorithm is compared with stylized reference benchmarks from optimal market making.

Figures

Figures reproduced from arXiv: 2601.17247 by the authors.

Figure 1
Figure 1. Episode 2000 for the benchmarks 6.3.1 Numerical method and parameter calibration In the first numerical implementation, we assume that the mid price St follows a rough Heston model. The motivation is based on the so-called rough volatility of financial assets [27, 12, 2]. Consider ρ ∈ [−1, 1] a constant, S0 = 100 (which is the numerical value we work with in this section), V0, H, θ, λ and ν be positive constants. Re… view at source ↗
Figure 2
Figure 2. Training analysis We now represents the behavior of the generative stochastic market model and the performance of the NFQ model over the last training episode [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Episode 2000 for the agent 22 [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

62 extracted references · 7 linked inside Pith

  1. [1]

    Lifting the Heston model.Quantitative finance, 19(12):1995–2013, 2019

    Eduardo Abi Jaber. Lifting the Heston model.Quantitative finance, 19(12):1995–2013, 2019

    1995

  2. [2]

    Multifactor approximation of rough volatility models.SIAM journal on financial mathematics, 10(2):309–349, 2019

    Eduardo Abi Jaber and Omar El Euch. Multifactor approximation of rough volatility models.SIAM journal on financial mathematics, 10(2):309–349, 2019

    2019

  3. [3]

    Optimal execution of portfolio transactions.Journal of Risk, 3:5–40, 2001

    Robert Almgren and Neil Chriss. Optimal execution of portfolio transactions.Journal of Risk, 3:5–40, 2001

    2001

  4. [4]

    High-frequency trading in a limit order book.Quan- titative Finance, 8(3):217–224, 2008

    Marco Avellaneda and Sasha Stoikov. High-frequency trading in a limit order book.Quan- titative Finance, 8(3):217–224, 2008

    2008

  5. [5]

    Algorithmic market making for options.Quantitative Finance, 21(1):85–97, 2021

    Bastien Baldacci, Philippe Bergault, and Olivier Guéant. Algorithmic market making for options.Quantitative Finance, 21(1):85–97, 2021

    2021

  6. [6]

    Market liq- uidity and competition among designated market makers.Management Science, 71(1):184– 201, 2025

    Mario Bellia, Loriana Pelizzon, Marti G Subrahmanyam, and Darya Yuferova. Market liq- uidity and competition among designated market makers.Management Science, 71(1):184– 201, 2025

    2025

  7. [7]

    Market making via reinforcement learning

    Taweh Beysolow II. Market making via reinforcement learning. InApplied Reinforcement Learning with Python: With OpenAI Gym, Tensorflow, and Keras, pages 77–94. Springer, 2019

    2019

  8. [8]

    IPO auctions: English, dutch,

    Bruno Biais and Anne Marie Faugeron-Crouzet. IPO auctions: English, dutch,... french, and internet.Journal of Financial Intermediation, 11(1):9–36, 2002

    2002

Show all 62 references
  1. [9]

    Price discovery and learning during the preopening period in the paris bourse.Journal of Political Economy, 107(6):1218–1248, 1999

    Bruno Biais, Pierre Hillion, and Chester Spatt. Price discovery and learning during the preopening period in the paris bourse.Journal of Political Economy, 107(6):1218–1248, 1999

    1999

  2. [10]

    Power laws in economics and finance: some ideas fromphysics

    Jean-Philippe Bouchaud. Power laws in economics and finance: some ideas fromphysics. Quantitative finance, 1(1):105, 2001

    2001

  3. [11]

    Implementation details for frequent batch auctions: Slowing down markets to the blink of an eye.American Economic Review, 104(5):418–424, 2014

    Eric Budish, Peter Cramton, and John Shim. Implementation details for frequent batch auctions: Slowing down markets to the blink of an eye.American Economic Review, 104(5):418–424, 2014

    2014

  4. [12]

    Portfolio optimization in fractional and rough Heston models.SIAM Journal on Financial Mathematics, 11(1):240–273, 2020

    Nicole Bäuerle and Sascha Desmettre. Portfolio optimization in fractional and rough Heston models.SIAM Journal on Financial Mathematics, 11(1):240–273, 2020

    2020

  5. [13]

    Logarithmic regret in the ergodic Avellaneda-Stoikov market making model.arXiv preprint arXiv:2409.02025, 2024

    Jialun Cao, David Šiška, Lukasz Szpruch, and Tanut Treetanthiploet. Logarithmic regret in the ergodic Avellaneda-Stoikov market making model.arXiv preprint arXiv:2409.02025, 2024

    2024 arXiv

  6. [14]

    Cambridge University Press, 2023

    Agostino Capponi and Charles-Albert Lehalle.Machine Learning and Data Sciences for Financial Markets: A Guide to Contemporary Practices. Cambridge University Press, 2023

    2023

  7. [15]

    The influence of economic research on financial mathematics: Evidence from the last 25 years.Finance and Stochastics, 26(1):85–101, 2022

    René Carmona. The influence of economic research on financial mathematics: Evidence from the last 25 years.Finance and Stochastics, 26(1):85–101, 2022

    2022

  8. [16]

    Cambridge University Press, 2015

    Álvaro Cartea, Sebastian Jaimungal, and José Penalva.Algorithmic and high-frequency trading. Cambridge University Press, 2015

    2015

  9. [17]

    Market making without regret.arXiv preprint arXiv:2411.13993, 2024

    Nicolò Cesa-Bianchi, Tommaso Cesari, Roberto Colomboni, Luigi Foscari, and Vinayak Pathak. Market making without regret.arXiv preprint arXiv:2411.13993, 2024. 25

    2024 arXiv

  10. [18]

    Special issue on machine learning in finance.Mathematical Finance, 34(2):259–261, 2024

    Christa Cuchiero, Hu Ruimeng, Sara Svaluto-Ferro, Xu Renyuan, et al. Special issue on machine learning in finance.Mathematical Finance, 34(2):259–261, 2024

    2024

  11. [19]

    AHEAD: Ad hoc electronic auction design.Frontiers of Mathematical Finance, 3(2):163–213, 2024

    Joffrey Derchu, Philippe Guillot, Thibaut Mastrolia, and Mathieu Rosenbaum. AHEAD: Ad hoc electronic auction design.Frontiers of Mathematical Finance, 3(2):163–213, 2024

    2024

  12. [20]

    Equilib- ria and incentives for illiquid auction markets.arXiv preprint arXiv:2307.15805, to appear in Market Microstructure and Liquidity, 2023

    JoffreyDerchu, DimitriosKavvathas, ThibautMastrolia, andMathieuRosenbaum. Equilib- ria and incentives for illiquid auction markets.arXiv preprint arXiv:2307.15805, to appear in Market Microstructure and Liquidity, 2023

    2023 arXiv

  13. [21]

    Welfare and optimal trading frequency in dynamic double auctions

    Songzi Du and Haoxiang Zhu. Welfare and optimal trading frequency in dynamic double auctions. Technical report, National Bureau of Economic Research, 2014

    2014

  14. [22]

    Size discovery.The Review of Financial Studies, 30(4):1095–1150, 2017

    Darrell Duffie and Haoxiang Zhu. Size discovery.The Review of Financial Studies, 30(4):1095–1150, 2017

    2017

  15. [23]

    A theoretical analysis of deep Q-learning

    Jianqing Fan, Zhaoran Wang, Yuchen Xie, and Zhuoran Yang. A theoretical analysis of deep Q-learning. InLearning for dynamics and control, pages 486–489. PMLR, 2020

    2020

  16. [24]

    Power laws in economics and finance.Annu

    Xavier Gabaix. Power laws in economics and finance.Annu. Rev. Econ., 1(1):255–294, 2009

    2009

  17. [25]

    Reinforcement learning for market making in a multi-agent dealer market.arXiv preprint arXiv:1911.05892, 2019

    Sumitra Ganesh, Nelson Vadori, Mengda Xu, Hua Zheng, Prashant Reddy, and Manuela Veloso. Reinforcement learning for market making in a multi-agent dealer market.arXiv preprint arXiv:1911.05892, 2019

    1911 arXiv

  18. [26]

    Market making with signals through deep rein- forcement learning.IEEE access, 9:61611–61622, 2021

    Bruno Gašperov and Zvonko Kostanjčar. Market making with signals through deep rein- forcement learning.IEEE access, 9:61611–61622, 2021

    2021

  19. [27]

    Volatility is rough

    Jim Gatheral, Thibault Jaisson, and Mathieu Rosenbaum. Volatility is rough. InCom- modities, pages 659–690. Chapman and Hall/CRC, 2022

    2022

  20. [28]

    Statis- tical properties of share volume traded in financial markets.Physical review e, 62(4):R4493, 2000

    Parameswaran Gopikrishnan, Vasiliki Plerou, Xavier Gabaix, and H Eugene Stanley. Statis- tical properties of share volume traded in financial markets.Physical review e, 62(4):R4493, 2000

    2000

  21. [29]

    Deepreinforcementlearningformarketmakingincorpo- rate bonds: beating the curse of dimensionality.Applied Mathematical Finance, 26(5):387– 452, 2019

    OlivierGuéantandIuliiaManziuk. Deepreinforcementlearningformarketmakingincorpo- rate bonds: beating the curse of dimensionality.Applied Mathematical Finance, 26(5):387– 452, 2019

    2019

  22. [30]

    Dealing with the inventory risk: a solution to the market making problem.Mathematics and financial eco- nomics, 7(4):477–507, 2013

    Olivier Guéant, Charles-Albert Lehalle, and Joaquin Fernandez-Tapia. Dealing with the inventory risk: a solution to the market making problem.Mathematics and financial eco- nomics, 7(4):477–507, 2013

    2013

  23. [31]

    Recent advances in reinforcement learning in finance.Mathematical Finance, 33(3):437–503, 2023

    Ben Hambly, Renyuan Xu, and Huining Yang. Recent advances in reinforcement learning in finance.Mathematical Finance, 33(3):437–503, 2023

    2023

  24. [32]

    Transaction cost (in) transparency: Coasian dynamics in frequent batch auctions.Available at SSRN 4861066, 2024

    Simon Jantschgi. Transaction cost (in) transparency: Coasian dynamics in frequent batch auctions.Available at SSRN 4861066, 2024

    2024

  25. [33]

    Market making with scaled beta poli- cies

    Joseph Jerome, Gregory Palmer, and Rahul Savani. Market making with scaled beta poli- cies. InProceedings of the Third ACM International Conference on AI in Finance, pages 214–222, 2022

    2022

  26. [34]

    Mbt- gym: Reinforcement learning for model-based limit order book trading

    Joseph Jerome, Leandro Sánchez-Betancourt, Rahul Savani, and Martin Herdegen. Mbt- gym: Reinforcement learning for model-based limit order book trading. InProceedings of the Fourth ACM International Conference on AI in Finance, pages 619–627, 2023. 26

    2023

  27. [35]

    The effect of a closing call auction on market quality and trading strategies.Journal of Financial Intermediation, 21(1):23–49, 2012

    Eugene Kandel, Barbara Rindi, and Luisella Bosetti. The effect of a closing call auction on market quality and trading strategies.Journal of Financial Intermediation, 21(1):23–49, 2012

    2012

  28. [36]

    Machine learning for market microstructure and high frequency trading.High frequency trading: New realities for traders, markets, and regulators, 72:1877–1901, 2013

    Michael Kearns and Yuriy Nevmyvaka. Machine learning for market microstructure and high frequency trading.High frequency trading: New realities for traders, markets, and regulators, 72:1877–1901, 2013

    1901

  29. [37]

    Continuous auctions and insider trading.Econometrica: Journal of the Econometric Society, pages 1315–1335, 1985

    Albert S Kyle. Continuous auctions and insider trading.Econometrica: Journal of the Econometric Society, pages 1315–1335, 1985

    1985

  30. [38]

    Learning a functional control for high- frequency finance.Quantitative Finance, 22(11):1973–1987, 2022

    Laura Leal, Mathieu Laurière, and C-A Lehalle. Learning a functional control for high- frequency finance.Quantitative Finance, 22(11):1973–1987, 2022

    1973

  31. [39]

    Trading mechanisms in securities markets.the Journal of Finance, 47(2):607–641, 1992

    Ananth Madhavan. Trading mechanisms in securities markets.the Journal of Finance, 47(2):607–641, 1992

    1992

  32. [40]

    Price discovery in auction markets: A look inside the black box.The Review of Financial Studies, 13(3):627–658, 2000

    Ananth Madhavan and Venkatesh Panchapagesan. Price discovery in auction markets: A look inside the black box.The Review of Financial Studies, 13(3):627–658, 2000

    2000

  33. [41]

    Clearing time randomization and transaction fees for auction market design.arXiv preprint arXiv:2405.09764, 2024

    Thibaut Mastrolia and Tianrui Xu. Clearing time randomization and transaction fees for auction market design.arXiv preprint arXiv:2405.09764, 2024

    2024 arXiv

  34. [42]

    Optimal rebate design: Incentives, competition and efficiency in auction markets.arXiv preprint arXiv:2501.12591, 2025

    Thibaut Mastrolia and Tianrui Xu. Optimal rebate design: Incentives, competition and efficiency in auction markets.arXiv preprint arXiv:2501.12591, 2025

    2025 arXiv

  35. [43]

    Auctions and bidding: A primer.Journal of economic perspectives, 3(3):3– 22, 1989

    Paul Milgrom. Auctions and bidding: A primer.Journal of economic perspectives, 3(3):3– 22, 1989

    1989

  36. [44]

    Auction market design: Recent innovations.Annual Review of Economics, 11(1):383–405, 2019

    Paul Milgrom. Auction market design: Recent innovations.Annual Review of Economics, 11(1):383–405, 2019

    2019

  37. [45]

    The economics of competitive bidding: a selective survey.Social goals and social organization: Essays in memory of Elisha Pazner, pages 261–292, 1985

    Paul R Milgrom. The economics of competitive bidding: a selective survey.Social goals and social organization: Essays in memory of Elisha Pazner, pages 261–292, 1985

    1985

  38. [46]

    Cambridge University Press, 2004

    Paul Robert Milgrom.Putting auction theory to work. Cambridge University Press, 2004

    2004

  39. [47]

    Playing atari with deep reinforcement learning.arXiv preprint arXiv:1312.5602, 2013

    Volodymyr Mnih. Playing atari with deep reinforcement learning.arXiv preprint arXiv:1312.5602, 2013

    2013 arXiv

  40. [48]

    Human-level control through deep reinforcement learning.nature, 518(7540):529–533, 2015

    Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, et al. Human-level control through deep reinforcement learning.nature, 518(7540):529–533, 2015

    2015

  41. [49]

    Enhancing Q-learning for optimal asset allocation.Advances in neural information processing systems, 10, 1997

    Ralph Neuneier. Enhancing Q-learning for optimal asset allocation.Advances in neural information processing systems, 10, 1997

    1997

  42. [50]

    Reinforcement learning for optimized trade execution

    Yuriy Nevmyvaka, Yi Feng, and Michael Kearns. Reinforcement learning for optimized trade execution. InProceedings of the 23rd international conference on Machine learning, pages 673–680, 2006

    2006

  43. [51]

    Double deep Q-learning for optimal execution.Applied Mathematical Finance, 28(4):361–380, 2021

    Brian Ning, Franco Ho Ting Lin, and Sebastian Jaimungal. Double deep Q-learning for optimal execution.Applied Mathematical Finance, 28(4):361–380, 2021

    2021

  44. [52]

    Optimal auction duration: A price formation viewpoint.Operations Research, 69(6):1734–1745, 2021

    Jusselin Paul, Mastrolia Thibaut, and Rosenbaum Mathieu. Optimal auction duration: A price formation viewpoint.Operations Research, 69(6):1734–1745, 2021. 27

    2021

  45. [53]

    John Wiley & Sons, 2014

    Martin L Puterman.Markov decision processes: discrete stochastic dynamic programming. John Wiley & Sons, 2014

    2014

  46. [54]

    The growing importance of the closing auction in share trading volumes

    Franck Raillon. The growing importance of the closing auction in share trading volumes. Journal of Securities Operations & Custody, 12(2):135–152, 2020

    2020

  47. [55]

    On the discrete-time simulation of the rough heston model.SIAM Journal on Financial Mathematics, 14(1):223–249, 2023

    Alexandre Richard, Xiaolu Tan, and Fan Yang. On the discrete-time simulation of the rough heston model.SIAM Journal on Financial Mathematics, 14(1):223–249, 2023

    2023

  48. [56]

    Neural fitted Q iteration–first experiences with a data efficient neural reinforcement learning method

    Martin Riedmiller. Neural fitted Q iteration–first experiences with a data efficient neural reinforcement learning method. InEuropean conference on machine learning, pages 317–

  49. [57]

    Equity auction dynamics: latent liquidity models with activity acceleration.Quantitative Finance, 24(10):1381–1398, 2024

    Mohammed Salek, Damien Challet, and Ioane Muni Toke. Equity auction dynamics: latent liquidity models with activity acceleration.Quantitative Finance, 24(10):1381–1398, 2024

    2024

  50. [58]

    Deep learning for limit order books.Quantitative Finance, 19(4):549– 570, 2019

    Justin A Sirignano. Deep learning for limit order books.Quantitative Finance, 19(4):549– 570, 2019

    2019

  51. [59]

    Richard S Sutton, Andrew G Barto, et al.Reinforcement learning: An introduction, vol- ume 1

  52. [60]

    Optimal dynamic auctions for revenue management.Management Science, 48(11):1388–1407, 2002

    Gustavo Vulcano, Garrett Van Ryzin, and Costis Maglaras. Optimal dynamic auctions for revenue management.Management Science, 48(11):1388–1407, 2002

    2002

  53. [61]

    Market making with learned beta policies

    Yongzhao Wang, Rahul Savani, Anri Gu, Chris Mascioli, Theodore Turocy, and Michael Wellman. Market making with learned beta policies. InProceedings of the 5th ACM International Conference on AI in Finance, pages 643–651, 2024

    2024

  54. [62]

    Q-learning.Machine learning, 8(3):279–292, 1992

    Christopher JCH Watkins and Peter Dayan. Q-learning.Machine learning, 8(3):279–292, 1992. 28

    1992

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.