Stochastic Control for Price-Responsive Optimal Execution
Summary
The document asks whether execution schedules can respond to prices observed during trading. It contrasts deterministic optimal execution approaches, which prescribe quantities without conditioning on realized prices, with stochastic control frameworks that can incorporate evolving market information. The cited research covers impulse control, liquidation with execution costs and risk, limit-order execution, and order books with controlled arrival intensity.
The response recommends stochastic control for making execution decisions responsive to observed conditions and notes that it can require more computation than deterministic control. It also points to Bayesian adaptive trading as an intermediate approach. The evidence consists of references to academic papers and a qualitative description of an illustration showing volume and mean trajectories under different trend conditions; no numerical performance comparison is supplied. The choice of framework therefore depends on how valuable price responsiveness is relative to modeling and computational complexity.
Key ideas
- Deterministic execution schedules do not condition trade quantities on prices observed during execution.
- Stochastic control can adapt trading decisions to evolving market conditions.
- Relevant formulations include impulse control, liquidation, limit-order execution, and controlled order arrival intensity.
- Bayesian adaptive trading is presented as another approach, while stochastic control may require more computation.
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Full text
# Optimal Executions for Minimizing Slippage # Optimal Executions for Minimizing Slippage There has been a considerable body of work for finding trading strategies that minimize the slippage wrt arrival price. For instance, the following are on of the most well known papers: [1] Robert Almgren, Neil Chriss, "Optimal execution of portfolio transactions" [2] Robert Almgren, "Optimal execution with nonlinear impact functions and trading-enhanced risk" [3] Mauricio Labadie, Charles-Albert Lehalle, "Optimal trading algorithms and self-similar processes: a p-variation approach" One criticism I have for these papers is the optimal trading quantities are independent on the price realization. More precisely, the number of shares to trade at time $t$ do not depends on the price I can observe at that moment which is an important piece of information. Is there any academic work where the price realization is incorporated in the decision process? ## Answer by lehalle (score 6, accepted) https://quant.stackexchange.com/a/10550 You are right, these work use deterministic control. Framework using stochastic control exist: - Bouchard, B., Dang, N.-M., Lehalle, C.-A., 2011. Optimal control of trading algorithms: a general impulse control approach. SIAM J. Financial Mathematics 2 (1), 404-438. URL http://epubs.siam.org/doi/abs/10.1137/090777293?af=R - Kharroubi, I., Pham, H., Jun. Optimal portfolio liquidation with execution cost and risk. SIAM J. Finan. Math., 1(1), 897–931. (35 pages) URL http://arxiv.org/abs/0906.2565 - Guéant, O., Lehalle, C.-A., Fernandez-Tapia, J., 2012. Optimal Execution with Limit Orders. SIAM Journal on Financial Mathematics 13 (1), 740-764. http://arxiv.org/abs/1106.3279 - Bayraktar, E., Ludkovski, M., Jun. 2012. Liquidation in limit order books with controlled intensity. Mathematical Finance. URL http://arxiv.org/abs/1105.0247 For market making you have few papers too. Below is a picture of the first paper in the list. You can see on the left the locally traded volume (in red) and on the right 3 mean trajectories (conditioned by the presence of a trend). It is a very good idea to use stochastic control; of course it is more CPU consuming than deterministic control. In between you have the Almgren and Lorenz paper: - Almgren, R., Lorenz, J., 2006. Bayesian adaptive trading with a daily cycle. Journal of Trading. http://www.courant.nyu.edu/~almgren/papers/drift.pdf
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