Optimal Execution as a Stochastic Control Problem

Summary

The article frames the execution of a large stock order as a tradeoff between market impact and exposure to price risk. Executing quickly may push the price against the trader, while slower execution leaves inventory exposed to uncertain price movements. It represents trading speed as a controllable process, alongside inventory, the stock’s midprice, execution price, and cash. A trader chooses a schedule to liquidate a target inventory by a specified horizon, with a terminal cost for shares left unsold.

The proposed solution is to define an expected objective over admissible trading speeds and derive a value function using the Hamilton-Jacobi-Bellman equation. This gives a framework for selecting an optimal liquidation rate under the model’s assumptions. The text indicates that a similar approach can be adapted to acquiring shares. However, the equations and resulting control rule are absent from the supplied document, so readers cannot reproduce the derivation or assess its parameters. It provides no market data, numerical example, or empirical comparison.

Key ideas

• Large orders create a tradeoff between price impact and the risk of waiting.
• The model treats trading speed as a control and inventory and cash as evolving state variables.
• A terminal penalty can represent the cost of inventory remaining at the execution horizon.
• The article proposes using a Hamilton-Jacobi-Bellman equation to derive an execution policy.
• The supplied text omits the equations and gives no empirical validation.

Tags

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