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Feasible Multi-Asset Optimal Execution under Cash Constraints

Ryuji Hashimoto, Namid R. Stillman

arXiv:2609.27786v1q-fin.TRcs.CEmath.OCq-fin.CPq-fin.RM

TL;DR

Multi-asset optimal execution research largely overlooks explicit cash limits along the execution path, leaving feasibility under limited capital poorly characterized. This paper adds per-period budgets on expected cumulative cash to the Almgren–Chriss framework. Tighter constraints shift schedules toward sell-first execution, and simulator evaluations report lower peak cash drawdown with comparable implementation shortfall.

  • Problem

    Existing multi-asset OE models largely overlook explicit cash constraints along execution or treat them heuristically, leaving feasibility under limited capital poorly characterized.

    Source: §1 · p. 1

  • Method

    The paper extends Almgren–Chriss with constraints requiring expected cumulative cash spent to remain within a prescribed budget at every trading period.

    Source: §1 · p. 1

  • Results

    The method substantially reduces peak cash drawdown while maintaining implementation shortfall comparable to unconstrained OE and standard benchmarks in out-of-sample agent-based simulations.

    Source: §1 · p. 1

  • Takeaways & Limitations

    Execution feasibility is an optimization objective alongside transaction cost and risk in practical multi-asset execution.

    Source: §7 · p. 8

    The framework uses classical linear market impact and constrains expected rather than realized cash trajectories, so stochastic prices may allow realized consumption to exceed the prescribed budget temporarily.

    Source: §7 · p. 8

Abstract

from arXiv · show

Optimal execution (OE) in multi-asset settings involves complex interactions across assets, particularly through shared capital constraints during portfolio rebalancing. While existing models capture cross-impact and portfolio-level dynamics, they largely overlook the role of explicit cash constraints along the execution trajectory. As a result, the feasibility of execution strategies under limited capital remains poorly understood. In this paper, we extend the classical Almgren-Chriss framework to incorporate intertemporal constraints on expected cash consumption, requiring that the expected cumulative cash spent does not exceed a prescribed budget at every trading period. We show that the resulting multi-asset OE problem can be equivalently formulated as a quadratically constrained quadratic program (QCQP), and further establish that it admits a convex representation under mild conditions. This provides a tractable framework for analyzing execution strategies under dynamic capital constraints. Through controlled synthetic experiments, we show that the proposed cash constraints qualitatively alter OE schedules toward cash-feasible sell-first executions as the constraints become tighter. Furthermore, evaluations in an out-of-sample agent-based market simulator demonstrate that our method substantially reduces peak cash drawdown while maintaining implementation shortfall comparable to existing execution strategies. Our results highlight the importance of explicitly modeling financial feasibility in multi-asset execution and provide a foundation for bridging theoretical OE models with practical capital constraints.

1 Introduction

The section identifies a gap in multi-asset optimal execution: explicit cash feasibility along the execution path is often absent or treated heuristically. It adds per-period expected-cash budgets and reports feasible, efficient execution strategies in synthetic and simulator experiments.

1 Introduction

  • Multi-asset OE must account for cross-impact and covariance structures [15], building on the Almgren–Chriss cost–risk framework and later extensions.Data-driven execution strategies have also been proposed.Source: §1 · p. 1
  • Existing multi-asset OE studies largely overlook cash constraints or treat them heuristically without a tractable analytical formulation.Early buy execution can deplete cash while a portfolio remains unbalanced, leaving the remaining plan financially and risk-wise infeasible.Source: §1 · p. 1
  • The extended Almgren–Chriss framework imposes expected cumulative-cash budgets at every trading period and yields an equivalent QCQP with a convex representation under mild conditions.Source: §1 · p. 1
  • Out-of-sample agent-based simulations substantially reduce peak cash drawdown while maintaining implementation shortfall comparable to unconstrained OE and standard benchmark strategies.The experiments also show tighter constraints postpone buys and advance sells, aligning cash inflows with outflows.Source: §1 · p. 1
  • Tighter cash constraints progressively postpone buy orders and advance sell orders in controlled experiments, aligning cash inflows with outflows.Source: §1 · p. 1

2 Related Work

Related work develops optimal execution from the Almgren–Chriss framework toward richer market dynamics and data-driven policies, and extends it to interacting assets. Explicit cash feasibility remains a limited analytical treatment in multi-asset models.

2 Related Work

  • The Almgren–Chriss framework models temporary and permanent market impact linearly and formulates execution as mean–variance optimization of implementation shortfall.Source: §2 · p. 1
  • AC linear-impact modeling was extended with order types, order flow, dynamic impact, and liquidity resilience [2, 4, 17].Source: §2 · pp. 1–2
  • The figure’s two-asset example moves from a 50–50 allocation toward 90% stock A and 10% stock B; buying first requires a cash outlay of 20% of initial portfolio value at t=1.The intermediate state also has equivalent excess risk exposure.Source: Figure 1 · p. 2
  • Data-driven approaches, including reinforcement learning, learn execution policies directly from market data in complex environments [18].Source: §2 · p. 2
  • Multi-asset models capture covariance and stochastic resilience [12] and cross-impact [1] [15], but typically omit cash constraints; Fang et al. use data-driven black-box policies.Source: §2 · p. 2

3 Problem Setup

The setup defines a discretized multi-asset execution problem with signed trade targets, AC-type price dynamics, and a mean–variance implementation-shortfall objective. Expected cumulative cash is limited at each time step.

Multi-Asset Optimal Execution Settings

  • The problem spans N assets with signed target quantities: positive quantities indicate purchases and negative quantities indicate sales.Source: §3 · p. 2
  • The finite horizon is divided into K equal-length trading periods, with orders submitted for each asset at each time step.Source: §3 · p. 2

IS and Cash Spent

  • Implementation shortfall is measured relative to initial reference prices; effective execution prices incorporate market impact and prevailing conditions, while cumulative cash records money spent through each time step.Source: §3 · p. 2

Price Process

  • The model assumes AC-type discrete-time price dynamics and linear market-impact functions with nonnegative impact coefficients.Source: §3 · p. 2

Safe OE Problem

  • The safe OE problem minimizes mean–variance implementation shortfall while matching each asset’s target and keeping expected cumulative cash within its budget at every time step.Source: §3 · pp. 2–3

4 Optimal Execution as a QCQP

The section recasts multi-asset optimal execution with dynamic cash limits as a QCQP and establishes convexity under nonnegative risk and impact parameters. Its derivation expresses the objective, inventory requirements, and expected cash constraints in quadratic or linear form.

4.1 Overview of the Main Claim

  • Under linear market-impact assumptions, the safe multi-asset execution problem is equivalent to a QCQP that can be handled with convex optimization techniques.The formulation is convex when the risk-aversion parameter and each asset’s temporary and permanent impact coefficients are nonnegative.Source: §4.1 · p. 3

4.2 Derivation of the QCQP Reformulation

  • The derivation expresses the execution objective and cumulative expected cash limits as quadratic forms in the schedule, while inventory targets become linear equalities.Together these components yield the QCQP representation of the constrained execution problem.Source: §4.2 · p. 3
  • The QCQP is convex when risk aversion and all assets’ temporary and permanent impact coefficients are nonnegative.Under these conditions, the objective and inequality constraints are convex, and the inventory equalities are affine.Source: §4.2 · p. 4

5 Experimental Setup

The experiments separate schedule-mechanism validation in an oracle setting from out-of-sample performance evaluation in a calibrated agent-based simulator. The setup specifies the cash-constraint variation, comparison strategies, evaluation metrics, and parameter-estimation protocol.

5 Experimental Setup

  • The study separates qualitative schedule analysis from practical performance evaluation using an oracle experiment and an out-of-sample agent-based market simulation.The first assumes known market parameters; the second estimates them from historical simulator data and evaluates schedules under out-of-sample conditions.Source: §5 · p. 4, §5.2 · p. 5

5.1 Experiment 1: Mechanism Validation

  • The mechanism experiment varies β in the budget path c̄_k = c̄_0 + βk for a two-asset portfolio that buys one asset and sells the other.All market parameters are assumed known to isolate the cash constraints’ qualitative effect on the schedule.Source: §5.1 · p. 4

5.2 Experiment 2: Performance Evaluation

  • Execution parameters are calibrated on separate trading days from evaluation, then held fixed during validation in a distinct market environment.Market-impact coefficients are estimated by OLS from TWAP runs; volatility is estimated from one-period mid-price returns in baseline simulations.Source: §5.2 · p. 5
  • The performance experiment compares the cash-constrained method with TWAP, VWAP, and an unconstrained ablation, reporting peak cash drawdown and combined portfolio implementation shortfall.The ablation removes the cash constraints while keeping the objective and other settings identical.Source: §5.2 · p. 5

6 Results

Tighter cash constraints reshape schedules toward earlier sells and later buys, coordinating cash inflows with outflows. In realistic-market simulations, the method substantially reduces peak cash drawdown while maintaining lower mean combined IS than AC free across evaluated asset pairs.

6.1 RQ1: Effect of Cash Constraints on OE

  • As the cash constraint tightens, buys are progressively postponed and sells advanced; at β = 0.1, most sells finish before substantial buys begin.Earlier sales generate expected inflows that finance later purchases while keeping expected cash within budget at every trading period.Source: §6.1 · p. 6

6.2 RQ2: Performance in Realistic Markets

  • The realistic-market evaluation uses calibrated parameters that vary across assets and validation-date price trajectories comprising multiple Monte Carlo realizations.The estimated market-impact and volatility parameters represent heterogeneous liquidity and price conditions.Source: §6.2 · p. 6, Figure 3 · p. 6
  • Across six ordered asset pairs, our method's peak cash drawdown ranges from 0.03% to 11.81%, versus 57.06% to 88.45% for AC free.Table 4 reports peak cash drawdown and combined IS as mean with standard deviation over 300 independent simulations, with Q(1) = 0.5 × 10^6 and Q(2) = −0.5 × 10^6.Source: §6.2 · p. 7, Table 4 · p. 7
  • At the benchmark volume, Ours has lower mean combined IS than AC free for every asset pair, but does not uniformly outperform TWAP and VWAP on IS.TWAP and VWAP attain lower combined IS for some pairs, while Ours performs best for others.Source: §6.2 · p. 7
  • Across 300 paired observations, Figure 4 shows lower peak cash drawdown for Ours and a mean combined-IS difference (Ours − AC free) of −6.68 bps (standard deviation: 12.95).The combined-IS distributions partially overlap across execution volumes and asset pairs.Source: Figure 4 · p. 7, §6.2 · p. 7

7 Discussion

Cash feasibility adds an execution objective alongside cost and risk: simultaneous buys and sells can create temporary shortages even when the target portfolio is financially feasible. The discussion highlights tractability and practical controls while identifying boundaries in the model and evaluation.

7 Discussion

  • Temporary cash shortages can arise during portfolio rebalancing with simultaneous buys and sells, even when the target portfolio is financially feasible.The classical focus on implementation shortfall, market impact, and execution risk implicitly assumes sufficient capital throughout execution.Source: §7 · p. 7
  • Cash constraints substantially reduce intermediate funding requirements while preserving execution efficiency, supporting feasibility as an objective alongside cost and risk.The formulation limits expected cumulative cash consumption at every execution period.Source: §7 · p. 8
  • Under standard linear market impact assumptions, the optimization remains a convex QCQP that existing convex solvers can solve for globally optimal schedules.This preserves computational tractability despite temporal dependencies across assets from cumulative cash flows.Source: §7 · p. 8
  • Practitioners can specify capital limits directly and retain interpretable schedules aligned with institutional funding and risk-management policies.The discussion presents this transparency as useful for institutional portfolio rebalancing.Source: §7 · p. 8
  • The framework assumes linear market impact and constrains expected, not realized, cash; empirical evaluation covers two-asset tasks calibrated from an agent-based simulator.Realized cash may exceed budget under stochastic prices, and larger-portfolio validation is still needed.Source: §7 · p. 8

8 Conclusion

The conclusion presents intertemporal expected-cash constraints as an extension of the classical Almgren–Chriss framework. The resulting convex QCQP reshapes schedules and reduces peak cash drawdown while maintaining comparable implementation shortfall in out-of-sample simulations.

8 Conclusion

  • Out-of-sample agent-based simulations show substantially lower peak cash drawdown while maintaining comparable implementation shortfall under various market conditions.The conclusion identifies feasibility-aware execution as a potential benchmark for future work on richer impact models, stochastic cash constraints, and learning-based execution.Source: §8 · p. 8
  • The multi-asset execution problem is reformulated as a convex QCQP, enabling efficient computation of globally optimal schedules while constraining expected capital deployment throughout execution.The formulation extends the classical Almgren–Chriss framework with intertemporal expected-cash constraints.Source: §8 · p. 8
  • Synthetic experiments show that cash constraints reshape schedules by coordinating cash inflows and outflows across assets.The conclusion describes this as a fundamental change in execution schedules.Source: §8 · p. 8
  • The authors argue that execution feasibility should be treated as a fundamental objective alongside execution cost and risk.They position the formulation as a benchmark for future feasibility-aware optimal execution studies.Source: §8 · p. 8

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