AlmgrenChriss

Almgren-Chriss refers to a framework for optimizing the execution of large trades to minimize execution costs and risk. Developed by Robert Almgren and Neil Chriss in the early 2000s, it provides a mathematical model for trading a specified quantity of an asset over a fixed horizon. The framework is widely regarded as a foundational approach in algorithmic and programmatic trading.

The model treats price dynamics as being affected by both permanent and temporary market impact from trading. The trader selects an execution rate v(t), the speed at which shares are sold or bought, over time. The asset price P(t) evolves with a stochastic component (capturing market risk) and a deterministic component reflecting permanent impact from cumulative trading, along with a term for temporary impact that depends on the instantaneous trading rate. The objective is to minimize the expected total cost of execution plus a risk term that grows with the variance of the execution cost. A risk aversion parameter governs the trade-off between cost minimization and risk containment.

Solutions in continuous time yield deterministic optimal trading trajectories under given parameters. In simple cases these trajectories can be explicit and smooth, often describing a front-loaded, evenly distributed, or other time-varying schedule depending on the balance between market impact and risk. Discrete-time implementations provide practical schedules used in real trading systems.

Almgren-Chriss remains a standard reference point for optimal execution. It has inspired numerous extensions, including multi-asset and portfolio contexts, non-linear impact, stochastic volatility, and additional market frictions, while also facing critique regarding its assumptions such as linear impact and Gaussian price dynamics.