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Learning Market Making with Closing Auctions

Julius Graf, Thibaut Mastrolia

TL;DR

This paper tackles market making across a trading session that ends with a closing auction, a key liquidity and price-discovery event often neglected by traditional models. It introduces a reinforcement-learning framework based on neural-fitted Q-learning that explicitly anticipates the closing auction, supported by a generative stochastic market model for simulation. The authors derive a general auction-clearing mechanism and demonstrate how a linear agent supply curve yields a closed-form solution, then validate the approach on both rough-Heston synthetic data and real SP500 assets, where the NFQ policy outperforms Avellaneda–Stoikov and TWAP baselines. The results highlight the practical value of auction-aware market making for improved liquidation, price discovery, and PnL across diverse market regimes.

Abstract

In this work, we investigate the market-making 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 Q-Learning framework 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 Deep Q-Learning method is 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 classical benchmarks from optimal market making.

Learning Market Making with Closing Auctions

TL;DR

This paper tackles market making across a trading session that ends with a closing auction, a key liquidity and price-discovery event often neglected by traditional models. It introduces a reinforcement-learning framework based on neural-fitted Q-learning that explicitly anticipates the closing auction, supported by a generative stochastic market model for simulation. The authors derive a general auction-clearing mechanism and demonstrate how a linear agent supply curve yields a closed-form solution, then validate the approach on both rough-Heston synthetic data and real SP500 assets, where the NFQ policy outperforms Avellaneda–Stoikov and TWAP baselines. The results highlight the practical value of auction-aware market making for improved liquidation, price discovery, and PnL across diverse market regimes.

Abstract

In this work, we investigate the market-making 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 Q-Learning framework 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 Deep Q-Learning method is 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 classical benchmarks from optimal market making.
Paper Structure (29 sections, 5 theorems, 33 equations, 3 figures, 4 tables, 2 algorithms)

This paper contains 29 sections, 5 theorems, 33 equations, 3 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Reinforcement learning and market making
  3. On the importance of (closing) auctions
  4. Methodology, contributions and financial insights
  5. Market model
  6. Mathematical framework and trading phases
  7. Trading during the continuous phase
  8. Trading during the auction phase
  9. Clearing price rule and estimation along the auction
  10. Projected hypothetical clearing price during the continuous session
  11. Markov Decision Process and dynamic programming for optimal market making with closing auction
  12. State space
  13. Action space.
  14. Reward.
  15. Learning market making with closing auction in an unknown environment
  16. ...and 14 more sections

Key Result

Theorem 2.1

Let $t_j \in \{t_{n+2},\ldots,t_m\}\cup\{\tau^{\mathrm{cl}}\}$. Assume that $\lim\limits_{p\to \pm\infty} g_{i,t_{j-1}}(p)=\pm\infty$ and $p\longmapsto g_{i,t_j}(p)$ is continuous and increasing for any $i\leq M_{t_{j-1}}$. Assume moreover that one of the following condition is satisfied Then, the estimated clearing price equation eq:hyp_cl_auction_estimation admits a unique solution.

Figures (3)

  • Figure 1: Episode 2000 for the benchmarks
  • Figure 2: Training analysis
  • Figure 3: Episode 2000 for the agent

Theorems & Definitions (25)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.1: Existence of a unique (estimated) clearing price
  • proof : Proof of Theorem \ref{['th:clearing']}
  • Corollary 2.1
  • proof
  • Remark 2.3
  • Proposition 2.1: Linear supply curve
  • Remark 2.4
  • Remark 3.1
  • ...and 15 more