Optimal Execution

Almgren & Chriss (2001) · Mean-Variance Execution Frontier

Family 07ExecutionStochastic control

Liquidating a large block of stock rapidly moves the price against you; liquidating it slowly exposes you to price drift. Almgren & Chriss (2001) formalised the trade-off: minimise expected execution cost $\mathbb{E}[C]$ plus $\lambda \cdot \text{Var}[C]$, under linear temporary and permanent market-impact models. The closed-form solution is an exponential trading schedule: $\lambda = 0$ gives a linear (VWAP-like) liquidation; higher $\lambda$ accelerates early, front-loading.

Educational Purpose

Educational demonstration of execution scheduling. Not trading advice. Production execution algorithms account for stochastic volatility, discrete tick sizes, venue fragmentation, adverse selection, and dark-pool dynamics far beyond the Almgren-Chriss base model.

The model

OR family: Trading & execution Solver class: Quadratic control Measure: Physical $\mathbb{P}$ Realism: ★★★ Closed-form

Setup

Trader holds $X_0$ shares to sell over $[0, T]$; at time $t_k = k\Delta t$ for $k = 1, \ldots, N$, they hold $x_k$ shares, having sold $n_k = x_{k-1} - x_k$ during the previous interval. Terminal constraint $x_N = 0$.

Impact model

Execution-price model (Almgren & Chriss 2001) $$ \tilde{S}_k \;=\; S_{k-1} - \gamma \, n_k - \tilde\eta \, v_k \;+\; \sigma \, \sqrt{\Delta t}\, \epsilon_k $$ $$ v_k \;=\; n_k / \Delta t \text{ is the trading rate} $$

$\gamma$ = permanent impact; $\tilde\eta$ = temporary impact; $\sigma$ = price volatility; $\epsilon_k \sim \mathcal{N}(0,1)$.

Mean-variance objective

Expected and variance of implementation shortfall $$ \mathbb{E}[C] \;=\; \tfrac{1}{2} \gamma X_0^2 + \tilde\eta \sum_{k=1}^{N} \frac{n_k^2}{\Delta t} $$ $$ \text{Var}[C] \;=\; \sigma^2 \sum_{k=1}^{N} x_k^2 \, \Delta t $$

Implementation shortfall = execution price minus mid-quote at decision time. Permanent-impact term is fixed at $\tfrac{1}{2}\gamma X_0^2$ regardless of schedule.

Closed-form optimal schedule

Minimising $\mathbb{E}[C] + \lambda \text{Var}[C]$ over $\{n_k\}$ yields an exponentially decaying holding:

Almgren-Chriss optimal trajectory $$ x_k \;=\; X_0 \cdot \frac{\sinh\!\bigl(\kappa (T - t_k)\bigr)}{\sinh(\kappa T)} $$ $$ \kappa \;=\; \sqrt{\frac{\lambda\, \sigma^2}{\tilde\eta}} $$

$\kappa = 0$ (i.e. $\lambda = 0$) gives the TWAP/linear schedule $x_k = X_0 (1 - t_k/T)$. Larger $\kappa$ accelerates early trading to reduce exposure to price drift.

Execution frontier

Varying $\lambda$ traces out the efficient execution frontier in the $(\text{Var}[C], \mathbb{E}[C])$ plane. Low $\lambda$ = cheap-but-risky (slow, exposed to drift); high $\lambda$ = expensive-but-safe (fast, high impact). Analogous to Markowitz’s efficient frontier in $(\sigma, \mu)$ space.

Interactive schedule

Trading-schedule solver

Shares $X_0$

1.00M

Horizon $T$ (hr)

3.0h

Steps $N$

Temp. impact $\eta$

1.0

Perm. impact $\gamma$

0.5

Volatility $\sigma$

2.0%

Risk aversion $\lambda$

2.0

$\mathbb{E}[C]$ (bps)

—

$\text{SD}[C]$ (bps)

—

$\kappa$

—

Half-life (hr)

—

Inventory $x_k$ vs time. Dashed gold = linear (VWAP, $\lambda = 0$). Solid blue = Almgren-Chriss exponential schedule at current $\lambda$. Higher $\lambda$ bends the curve below linear (trade earlier).

Execution frontier: $\mathbb{E}[C]$ vs $\text{SD}[C]$ as $\lambda$ varies. Current choice highlighted.

Extensions & limitations

Model extensions

Nonlinear impact. Empirically, impact scales as $v^\beta$ with $\beta \approx 0.5$ (square-root law); Gatheral (2010) derives no-arbitrage constraints on impact-decay kernels.
Multi-asset execution. Cross-impact between correlated assets; closed-form extends to matrix-valued $\eta, \gamma$.
Stochastic volatility / liquidity. GARCH-type volatility and time-varying liquidity require HJB-PDE or reinforcement-learning solutions.
Benchmark-relative execution. Minimise tracking error to VWAP / TWAP / Arrival-Price benchmarks (Kissell 2013).
Dark pools & fragmentation. Order routing across dozens of venues with different fee / rebate / fill-probability profiles.

Limitations

Linear impact assumption is first-order only; square-root fits empirical data better.
Brownian-motion price is a simplification; real prices exhibit jumps and regime changes.
Trader has no information advantage and no adverse selection concerns in the base model.
Fixed parameters $\eta, \gamma, \sigma$ are estimated; misspecification leads to sub-optimal schedules.

Key references

Almgren, R. & Chriss, N. (2001).

Optimal Execution of Portfolio Transactions.

Journal of Risk, 3, 5–40. doi:10.21314/JOR.2001.041

Bertsimas, D. & Lo, A. W. (1998).

Optimal Control of Execution Costs.

Journal of Financial Markets, 1(1), 1–50. doi:10.1016/S1386-4181(97)00012-8

Gatheral, J. (2010).

No-Dynamic-Arbitrage and Market Impact.

Quantitative Finance, 10(7), 749–759. doi:10.1080/14697680903373692

Cartea, Á., Jaim, S. & Penalva, J. (2015).

Algorithmic and High-Frequency Trading.

Cambridge University Press. ISBN 978-1-107-09114-6.

Guéant, O. (2016).

The Financial Mathematics of Market Liquidity: From Optimal Execution to Market Making.

CRC Press. ISBN 978-1-4987-2547-7.

Kissell, R. (2013).

The Science of Algorithmic Trading and Portfolio Management.

Academic Press. ISBN 978-0-12-401689-7.

Obizhaeva, A. A. & Wang, J. (2013).

Optimal Trading Strategy and Supply/Demand Dynamics.

Journal of Financial Markets, 16(1), 1–32. doi:10.1016/j.finmar.2012.09.001