Optimal Execution

High-Frequency Analysis of a Trading Game with Transient Price Impact ArXiv ID: 2512.11765 “View on arXiv” Authors: Marcel Nutz, Alessandro Prosperi Abstract We study the high-frequency limit of an $n$-trader optimal execution game in discrete time. Traders face transient price impact of Obizhaeva–Wang type in addition to quadratic instantaneous trading costs $θ(ΔX_t)^2$ on each transaction $ΔX_t$. There is a unique Nash equilibrium in which traders choose liquidation strategies minimizing expected execution costs. In the high-frequency limit where the grid of trading dates converges to the continuous interval $[“0,T”]$, the discrete equilibrium inventories converge at rate $1/N$ to the continuous-time equilibrium of an Obizhaeva–Wang model with additional quadratic costs $\vartheta_0(ΔX_0)^2$ and $\vartheta_T(ΔX_T)^2$ on initial and terminal block trades, where $\vartheta_0=(n-1)/2$ and $\vartheta_T=1/2$. The latter model was introduced by Campbell and Nutz as the limit of continuous-time equilibria with vanishing instantaneous costs. Our results extend and refine previous results of Schied, Strehle, and Zhang for the particular case $n=2$ where $\vartheta_0=\vartheta_T=1/2$. In particular, we show how the coefficients $\vartheta_0=(n-1)/2$ and $\vartheta_T=1/2$ arise endogenously in the high-frequency limit: the initial and terminal block costs of the continuous-time model are identified as the limits of the cumulative discrete instantaneous costs incurred over small neighborhoods of $0$ and $T$, respectively, and these limits are independent of $θ>0$. By contrast, when $θ=0$ the discrete-time equilibrium strategies and costs exhibit persistent oscillations and admit no high-frequency limit, mirroring the non-existence of continuous-time equilibria without boundary block costs. Our results show that two different types of trading frictions – a fine time discretization and small instantaneous costs in continuous time – have similar regularizing effects and select a canonical model in the limit. ...

Reinforcement Learning in Queue-Reactive Models: Application to Optimal Execution ArXiv ID: 2511.15262 “View on arXiv” Authors: Tomas Espana, Yadh Hafsi, Fabrizio Lillo, Edoardo Vittori Abstract We investigate the use of Reinforcement Learning for the optimal execution of meta-orders, where the objective is to execute incrementally large orders while minimizing implementation shortfall and market impact over an extended period of time. Departing from traditional parametric approaches to price dynamics and impact modeling, we adopt a model-free, data-driven framework. Since policy optimization requires counterfactual feedback that historical data cannot provide, we employ the Queue-Reactive Model to generate realistic and tractable limit order book simulations that encompass transient price impact, and nonlinear and dynamic order flow responses. Methodologically, we train a Double Deep Q-Network agent on a state space comprising time, inventory, price, and depth variables, and evaluate its performance against established benchmarks. Numerical simulation results show that the agent learns a policy that is both strategic and tactical, adapting effectively to order book conditions and outperforming standard approaches across multiple training configurations. These findings provide strong evidence that model-free Reinforcement Learning can yield adaptive and robust solutions to the optimal execution problem. ...

Optimal Execution under Liquidity Uncertainty ArXiv ID: 2506.11813 “View on arXiv” Authors: Etienne Chevalier, Yadh Hafsi, Vathana Ly Vath, Sergio Pulido Abstract We study an optimal execution strategy for purchasing a large block of shares over a fixed time horizon. The execution problem is subject to a general price impact that gradually dissipates due to market resilience. This resilience is modeled through a potentially arbitrary limit-order book shape. To account for liquidity dynamics, we introduce a stochastic volume effect governing the recovery of the deviation process, which represents the difference between the impacted and unaffected price. Additionally, we incorporate stochastic liquidity variations through a regime-switching Markov chain to capture abrupt shifts in market conditions. We study this singular control problem, where the trader optimally determines the timing and rate of purchases to minimize execution costs. The associated value function to this optimization problem is shown to satisfy a system of variational Hamilton-Jacobi-Bellman inequalities. Moreover, we establish that it is the unique viscosity solution to this HJB system and study the analytical properties of the free boundary separating the execution and continuation regions. To illustrate our results, we present numerical examples under different limit-order book configurations, highlighting the interplay between price impact, resilience dynamics, and stochastic liquidity regimes in shaping the optimal execution strategy. ...

FlowOE: Imitation Learning with Flow Policy from Ensemble RL Experts for Optimal Execution under Heston Volatility and Concave Market Impacts ArXiv ID: 2506.05755 “View on arXiv” Authors: Yang Li, Zhi Chen Abstract Optimal execution in financial markets refers to the process of strategically transacting a large volume of assets over a period to achieve the best possible outcome by balancing the trade-off between market impact costs and timing or volatility risks. Traditional optimal execution strategies, such as static Almgren-Chriss models, often prove suboptimal in dynamic financial markets. This paper propose flowOE, a novel imitation learning framework based on flow matching models, to address these limitations. FlowOE learns from a diverse set of expert traditional strategies and adaptively selects the most suitable expert behavior for prevailing market conditions. A key innovation is the incorporation of a refining loss function during the imitation process, enabling flowOE not only to mimic but also to improve upon the learned expert actions. To the best of our knowledge, this work is the first to apply flow matching models in a stochastic optimal execution problem. Empirical evaluations across various market conditions demonstrate that flowOE significantly outperforms both the specifically calibrated expert models and other traditional benchmarks, achieving higher profits with reduced risk. These results underscore the practical applicability and potential of flowOE to enhance adaptive optimal execution. ...

Shortermism and excessive risk taking in optimal execution with a target performance ArXiv ID: 2505.15611 “View on arXiv” Authors: Emilio Barucci, Yuheng Lan Abstract We deal with the optimal execution problem when the broker’s goal is to reach a performance barrier avoiding a downside barrier. The performance is provided by the wealth accumulated by trading in the market, the shares detained by the broker evaluated at the market price plus a slippage cost yielding a quadratic inventory cost. Over a short horizon, this type of remuneration leads, at the same time, to a more aggressive and less risky strategy compared to the classical one, and over a long horizon the performance turns to be poorer and more dispersed. ...

Optimal Execution in Intraday Energy Markets under Hawkes Processes with Transient Impact ArXiv ID: 2504.10282 “View on arXiv” Authors: Unknown Abstract This paper investigates optimal execution strategies in intraday energy markets through a mutually exciting Hawkes process model. Calibrated to data from the German intraday electricity market, the model effectively captures key empirical features, including intra-session volatility, distinct intraday market activity patterns, and the Samuelson effect as gate closure approaches. By integrating a transient price impact model with a bivariate Hawkes process to model the market order flow, we derive an optimal trading trajectory for energy companies managing large volumes, accounting for the specific trading patterns in these markets. A back-testing analysis compares the proposed strategy against standard benchmarks such as Time-Weighted Average Price (TWAP) and Volume-Weighted Average Price (VWAP), demonstrating substantial cost reductions across various hourly trading products in intraday energy markets. ...

Optimal Execution and Macroscopic Market Making ArXiv ID: 2504.06717 “View on arXiv” Authors: Unknown Abstract We propose a stochastic game modelling the strategic interaction between market makers and traders of optimal execution type. For traders, the permanent price impact commonly attributed to them is replaced by quoting strategies implemented by market makers. For market makers, order flows become endogenous, driven by tactical traders rather than assumed exogenously. Using the forward-backward stochastic differential equation (FBSDE) characterization of Nash equilibria, we establish a local well-posedness result for the general game. In the specific Almgren-Chriss-Avellaneda-Stoikov model, a decoupling approach guarantees the global well-posedness of the FBSDE system via the well-posedness of an associated backward stochastic Riccati equation. Finally, by introducing small diffusion terms into the inventory processes, global well-posedness is achieved for the approximation game. ...

Randomization in Optimal Execution Games ArXiv ID: 2503.08833 “View on arXiv” Authors: Unknown Abstract We study optimal execution in markets with transient price impact in a competitive setting with $N$ traders. Motivated by prior negative results on the existence of pure Nash equilibria, we consider randomized strategies for the traders and whether allowing such strategies can restore the existence of equilibria. We show that given a randomized strategy, there is a non-randomized strategy with strictly lower expected execution cost, and moreover this de-randomization can be achieved by a simple averaging procedure. As a consequence, Nash equilibria cannot contain randomized strategies, and non-existence of pure equilibria implies non-existence of randomized equilibria. Separately, we also establish uniqueness of equilibria. Both results hold in a general transaction cost model given by a strictly positive definite impact decay kernel and a convex trading cost. ...

Fredholm Approach to Nonlinear Propagator Models ArXiv ID: 2503.04323 “View on arXiv” Authors: Unknown Abstract We formulate and solve an optimal trading problem with alpha signals, where transactions induce a nonlinear transient price impact described by a general propagator model, including power-law decay. Using a variational approach, we demonstrate that the optimal trading strategy satisfies a nonlinear stochastic Fredholm equation with both forward and backward coefficients. We prove the existence and uniqueness of the solution under a monotonicity condition reflecting the nonlinearity of the price impact. Moreover, we derive an existence result for the optimal strategy beyond this condition when the underlying probability space is countable. In addition, we introduce a novel iterative scheme and establish its convergence to the optimal trading strategy. Finally, we provide a numerical implementation of the scheme that illustrates its convergence, stability, and the effects of concavity on optimal execution strategies under exponential and power-law decay. ...

To Hedge or Not to Hedge: Optimal Strategies for Stochastic Trade Flow Management ArXiv ID: 2503.02496 “View on arXiv” Authors: Unknown Abstract This paper addresses the trade-off between internalisation and externalisation in the management of stochastic trade flows. We consider agents who must absorb flows and manage risk by deciding whether to warehouse it or hedge in the market, thereby incurring transaction costs and market impact. Unlike market makers, these agents cannot skew their quotes to attract offsetting flows and deter risk-increasing ones, leading to a fundamentally different problem. Within the Almgren-Chriss framework, we derive almost-closed-form solutions in the case of quadratic execution costs, while more general cases require numerical methods. In particular, we discuss the challenges posed by artificial boundary conditions when using classical grid-based numerical PDE techniques and propose reinforcement learning methods as an alternative. ...