Optimal Stock Liquidation with a Stochastic Control Model

Summary

This article formulates the liquidation of a large stock position as a stochastic optimal control problem. The trader chooses a nonnegative trading speed that reduces inventory over time. Faster trading incurs greater price impact, modeled as a cost that grows with speed, while slower trading leaves more inventory exposed to market risk and a penalty for shares remaining at the terminal time.

The objective combines execution proceeds, terminal valuation of unliquidated shares, and a quadratic penalty for leftover inventory. The article states a Hamilton–Jacobi–Bellman equation for the value function and gives a constant optimal liquidation speed under its assumptions. This is a stylized model: it treats the midprice as Brownian, uses a simplified linear impact term, and does not provide calibration or empirical performance evidence. Its result should therefore be read as a theoretical illustration of the speed-versus-risk tradeoff, not a complete execution policy for real markets.

Key ideas

• The trader controls liquidation speed to balance execution impact against exposure to market risk.
• The model tracks inventory decline and assumes the midprice follows a Brownian process.
• Execution price is reduced by the bid-ask spread and a speed-dependent impact cost.
• A terminal quadratic penalty discourages leaving inventory unsold by the deadline.
• The article derives an optimal constant speed for its simplified model, without empirical validation.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.