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Almgren-Chriss Model

Algorithm description

The Almgren-Chriss model (2001) is the canonical static mean-variance framework for optimal execution of a large order. It splits impact into a permanent linear component and a temporary linear component, then minimises the Markowitz-style cost functional

\[\mathcal{C}(X) \;=\; \mathbb{E}[\text{execution cost}] \;+\; \lambda \cdot \text{Var}[\text{execution cost}]\]

over a deterministic trading schedule \(X_t\) on \([0, T]\) with \(X_0 = x\) (initial inventory) and \(X_T = 0\) (full liquidation).

The continuous-time solution is an exponentially-decaying trading rate:

\[X^*_t \;=\; x \cdot \frac{\sinh(\kappa (T - t))}{\sinh(\kappa T)}, \qquad \kappa = \sqrt{\lambda \sigma^2 / \eta}\]

where \(\sigma\) is volatility, \(\eta\) is the temporary-impact coefficient, and \(\lambda\) is the risk-aversion parameter. Rate \(dX^*_t / dt\) is highest near \(t = 0\) and \(t = T\), lowest in the middle — unless \(\lambda = 0\), in which case the optimum collapses to uniform trading (TWAP).

The cost model

Almgren-Chriss assumes two linear impact components:

  • Permanent impact \(g(\dot X_t) = \gamma \dot X_t\): a drift term that persists. Quadratic contribution to total cost via \(\int \dot X \cdot X \, dt\).
  • Temporary impact \(h(\dot X_t) = \eta \dot X_t\): a slippage on each instantaneous trade. Quadratic contribution via \(\int \eta \dot X_t^2 \, dt\).

Expected cost:

\[\mathbb{E}[\text{cost}] \;=\; \frac{\gamma}{2} x^2 \;+\; \eta \int_0^T \dot X_t^2 \, dt\]

Cost variance (from unhedged price risk on remaining inventory):

\[\text{Var}[\text{cost}] \;=\; \sigma^2 \int_0^T X_t^2 \, dt\]

The minimiser of \(\mathbb{E} + \lambda \text{Var}\) is the closed-form \(X^*_t\) above.

The efficient frontier

Varying \(\lambda\) traces out an efficient frontier analogous to Markowitz portfolio theory:

  • \(\lambda \to 0\): risk-neutral, schedule is TWAP (uniform trading), cost minimised but variance maximal.
  • \(\lambda \to \infty\): risk-averse, schedule front-loads aggressively to shrink exposure time, cost high but variance minimal.
  • Intermediate \(\lambda\): exponential-decay schedule trading off cost vs variance.

Practitioners pick a point on the frontier based on their mandate (cost-sensitive asset manager vs urgency-sensitive portfolio liquidator).

Variants and extensions

Variant Modification
Bertsimas-Lo (1998) Discrete-time DP; linear impact; no risk term → always TWAP. Precursor to AC.
Almgren-Chriss (2001) Adds quadratic risk term; introduces the efficient frontier.
Almgren (2003) Nonlinear (power-law) temporary impact; preserves closed-form family.
Obizhaeva-Wang (2013) Replaces the flat impact with transient impact (exponential decay kernel); schedule becomes singular (jumps at endpoints).
Gatheral-Schied-Slynko Generalises Obizhaeva-Wang to arbitrary positive-definite decay kernels. See propagator-model.
Cartea-Jaimungal Instantaneous impact + continuous adaptive trading with a Markovian signal.
Lehalle-Neuman Adds a Markovian signal to GSS. See signal-aware-optimal-execution.

The lineage is: Almgren-Chriss → Obizhaeva-Wang (add impact decay) → Lehalle-Neuman (add signal). Almgren-Chriss is the baseline every subsequent framework compares against.

Why it's the baseline

Every modern execution paper benchmarks against Almgren-Chriss because:

  1. Analytical tractability. Closed-form schedule, closed-form cost, closed-form variance. Easy to sanity-check numerical algorithms.
  2. Interpretability. The three parameters \((\sigma, \eta, \lambda)\) each map to a concrete intuition (volatility, temporary impact, risk aversion).
  3. Industry default. TWAP and VWAP algos are AC special cases (\(\lambda \to 0\) and \(\lambda \to 0\) with volume-weighting respectively). Any new algorithm needs to beat AC on either cost, variance, or robustness to non-stationarity.
  4. Pedagogical clarity. The mean-variance structure mirrors portfolio theory, making the tradeoff legible to anyone with a Markowitz background.

When to use / when not to

Use when: - You need a quick, interpretable baseline schedule. - Your market has roughly linear temporary impact and you're only trading a few tens of minutes. - You want the simplest possible risk-adjusted execution plan.

Don't use when: - Impact is strongly concave (square-root law dominates at larger sizes). - You have a short-horizon signal you want to exploit — use signal-aware-optimal-execution or MPC instead. - Market impact is transient with long memory — use propagator-model for correct self-impact accounting. - You need an adaptive online strategy — AC is a static schedule set once at \(t = 0\).

Limitations

  • Linear impact is a strong assumption. Empirical impact is concave (square-root law); AC underestimates cost for large orders.
  • Permanent-vs-temporary split is a modelling convention. Real impact decays on multiple timescales; AC collapses this into two numbers.
  • No signal awareness. The schedule doesn't react to market information. Any predictive edge is left on the table.
  • No transaction costs or fees. These are assumed folded into \(\eta\) but rarely dominate the shape of the solution.
  • No multi-venue routing. Modern fragmented markets require cross-venue decisions AC doesn't address.
  • Deterministic schedule. The realised path doesn't adapt to the actual fills — in practice, brokers re-plan periodically, which is an ad-hoc patch on AC's static nature.

Implementations

  • Industry: virtually every broker's execution desk ships an AC-based algorithm as the "urgency slider" control (\(\lambda\)) on implementation-shortfall schedulers.
  • Open source: QuantLib, pyfolio-execution, and many Python execution toolkits include AC as a reference.
  • In this wiki: referenced as baseline in mpc-trade-execution (MPC improves on AC), lehalle-neuman-signals-optimal-trading (AC is the no-signal limit), brokmann-slow-decay-impact (propagator-corrected AC cost estimates), and bouchaud-farmer-lillo-propagator (AC's permanent-impact component contradicts the empirical slow-decay story).

Connections