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Quant Systems Lab · Control Systems for Quantitative Finance
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Optimal Execution as Stochastic Control

Optimal execution balances market impact and risk over time, typically via quadratic cost control problems.

Explanation

Execution models trade off expected cost (impact) against risk (price uncertainty during the schedule).

Almgren–Chriss-type setups yield closed-form optimal schedules for simple impact and volatility assumptions.

In realistic systems, constraints, discrete time, and signals turn execution into a high-dimensional control problem.

executioncontrolimpactschedule
Interactive visualisation

Optimal execution balances market impact against price risk over time. Here a simple schedule is shaped by volatility, impact strength, and risk aversion: more risk aversion front-loads the trades, while stronger impact spreads them out.

Trade sizes q_t over the schedule (blue bars); larger λ front-loads, larger γ spreads outt1t3t5t7t9Σ q_t = 50.00
Impact cost vs risk for this schedule (normalised to compare)Impact cost270.47Risk3.04Normalised cost–risk score ≈ 0.60
Numbers
Σ q_t = 50.00, σ ≈ 20%
Impact γ ≈ 0.40, risk aversion λ ≈ 2.5
Expected impact cost ≈ 270.47
Risk measure ≈ 3.04
Cost–risk score ≈ 0.60
Interpretation

With low λ and strong impact, the schedule spreads trades more evenly to avoid large instantaneous blocks. As λ increases, the blue bars shift towards the start: you accept higher impact to reduce the time you carry risk.

The bottom panel shows the two sides of the trade-off. The aim of optimal execution is not to minimise either bar in isolation, but to choose a schedule that achieves an acceptable balance given your volatility, impact estimates, and risk appetite.