Optimal Execution and the Almgren-Chriss Model

Optimal execution is the problem of how to buy or sell a large position over time at the lowest total cost. Trade too fast and your orders push the price against you. Trade too slowly and the price may move away for other reasons while you wait. The most famous framework for this trade off is the model published by Robert Almgren and Neil Chriss in 2000, which still underpins many execution algorithms.

The trade off at the heart of execution#

There is no free option: lowering one cost raises the other. Optimal execution picks a schedule based on how much the trader dislikes uncertainty.

The Almgren-Chriss model in plain words#

The model makes a few simplifying assumptions:

1. You must trade a fixed number of shares within a fixed time.
2. Prices move randomly, with a known volatility.
3. Your trading creates temporary impact, which depends on how fast you trade at each moment, and permanent impact, which shifts the price by an amount proportional to how much you have traded.
4. You care about both the expected cost and the variance (uncertainty) of the cost, weighted by a risk aversion parameter.

The model then finds the trading schedule that minimises expected cost plus risk aversion times variance.

What the solution looks like#

• A risk neutral trader (who only cares about expected cost) trades at a constant rate, spreading the order evenly. This matches a time weighted (TWAP) schedule.
• A risk averse trader front loads the order, trading faster at the start to reduce exposure to price moves, then slowing down.
• Higher volatility pushes towards faster trading, because waiting is riskier.
• Higher impact costs push towards slower trading.

The efficient frontier of execution#

Plotting expected cost against the variance of cost for every possible schedule gives a curve similar to the efficient frontier in portfolio theory. Each point on the curve is the cheapest schedule for a given level of risk. The trader picks a point based on their risk tolerance. See Modern Portfolio Theory and the Efficient Frontier.

Beyond the basic model#

Real execution adds complications the original model leaves out:

• Intraday volume patterns: trading more when volume is naturally high reduces impact, the basis of VWAP algorithms. See VWAP, TWAP and POV Execution.
• Changing liquidity and spreads.
• Short term signals about price direction, which should speed up or slow down trading.
• Order types and venues: passive orders, dark pools and auctions.
• Non linear impact: empirical impact grows roughly with the square root of size. See Market Impact.

Modern execution algorithms combine these elements, often adapting in real time.

Why it matters even if you trade small#

Most individual traders never need an execution schedule. But the same logic applies to any decision to scale into a position: entering all at once accepts more immediate cost, while entering in pieces accepts the risk of the price moving. Understanding the trade off helps when building positions in thinner markets.

Frequently asked questions#

What is the Almgren-Chriss model?#

A model for executing large orders that chooses a trading schedule balancing market impact costs against the risk of price changes, based on the trader's risk aversion.

What is optimal execution?#

The practice of trading a large order in a way that minimises total cost, including impact, spread and the risk of unfavourable price moves.

Do retail traders need optimal execution?#

Rarely, because their orders are small. It matters for funds, large traders and anyone trading illiquid markets.

Sources#

• Almgren, R. and Chriss, N., Optimal execution of portfolio transactions, Journal of Risk, 2000
• Wikipedia, Algorithmic trading