Almgren-Chriss Optimal Execution

Setup

The Execution Problem

A portfolio manager needs to sell $X$ shares of a stock over a time horizon $[0, T]$. Selling too fast incurs large market impact costs (moving the price against themselves). Selling too slowly leaves the position exposed to price risk for longer. This is the optimal execution problem: find a trading schedule that balances market impact against timing risk.

This is a ubiquitous problem in practice: unwinding a large position post-trade decision, rebalancing a portfolio after a factor model update, or executing a delta hedge for a large options position. The Almgren-Chriss (2001) framework is the standard model for this problem on institutional trading desks.

Conventions and Assumptions

• Initial position: $X > 0$ shares to sell, $X_0 = X$.
• Time grid: discrete times $0 = t_0 < t_1 < \cdots < t_N = T$ with equal spacing $\tau = T/N$.
• Shares sold in period $k$: $n_k = X_{k-1} - X_k \geq 0$ (non-negative, selling only).
• Trade rate: $v_k = n_k / \tau$ (shares per unit time).
• Mid-price dynamics in the absence of trading: $S_k^0 = S_{k-1}^0 + \sigma\sqrt{\tau}\, \xi_k$ where $\xi_k \stackrel{\text{i.i.d.}}{\sim} \mathcal{N}(0,1)$.
• Volatility $\sigma$: annualised. Daily vol is $\sigma\sqrt{1/252}$ for equities.
• Rates are zero (P&L is measured in dollar terms, no discounting).

Market Impact Model

Permanent and Temporary Impact

Almgren-Chriss decompose market impact into two components:

Permanent impact: trades shift the mid-price permanently. Every share sold at time $k$ reduces the mid-price for all future periods. The cumulative permanent impact of selling $n_1, \ldots, n_k$ shares is:

$\Delta S_k^{\mathrm{perm}} = -g\!\left(\frac{n_k}{\tau}\right) \cdot \tau = -g(v_k)\cdot\tau,$

where $g(v)$ is the permanent impact function (shares per time → price change). For the linear model: $g(v) = \gamma v$, so selling at rate $v$ permanently moves the price by $-\gamma v \cdot \tau$ per period.

Temporary impact: the execution price in period $k$ differs from the (already impacted) mid by an additional amount due to consuming liquidity immediately. This cost is paid once and does not affect future mid-prices:

$P_k^{\mathrm{exec}} = \tilde{S}_k - h\!\left(\frac{n_k}{\tau}\right),$

where $\tilde{S}_k$ is the mid-price at time $k$ (already shifted by permanent impact), and $h(v)$ is the temporary impact function. For the linear model: $h(v) = \epsilon\, \mathrm{sgn}(v) + \eta v$, where $\epsilon$ is a fixed half-spread cost and $\eta v$ is the linear temporary impact.

The Full Price Process

Let $\tilde{S}_k$ be the mid-price after permanent impact. Starting from $S_0$:

$\tilde{S}_k = S_0 + \sigma\sqrt{\tau}\sum_{j=1}^k \xi_j - \gamma \sum_{j=1}^k n_j.$

The total proceeds from selling $X$ shares:

$\mathrm{Proceeds} = \sum_{k=1}^N n_k P_k^{\mathrm{exec}} = \sum_{k=1}^N n_k \!\left[\tilde{S}_k - h(n_k/\tau)\right].$

The shortfall (cost of execution relative to the decision price $S_0$):

$\mathrm{IS} = S_0 X - \mathrm{Proceeds} = \underbrace{\gamma \sum_{k=1}^N n_k \sum_{j=1}^k n_j}_{\mathrm{permanent\,impact}} + \underbrace{\eta \sum_{k=1}^N \frac{n_k^2}{\tau}}_{\mathrm{temporary\,impact}} + \underbrace{\sigma\sqrt{\tau}\sum_{k=1}^N n_k \sum_{j=1}^k \xi_j}_{\mathrm{timing\,risk}}.$

The first two terms are deterministic costs; the last is random (timing risk from mid-price moves while the position is open).

Theory: Optimal Execution

Objective Function

Almgren-Chriss optimise the mean-variance objective over the implementation shortfall:

$U = \mathbb{E}[\mathrm{IS}] + \frac{\lambda}{2} \mathrm{Var}[\mathrm{IS}],$

where $\lambda \geq 0$ is the risk-aversion parameter (units: inverse variance, or equivalently, the penalty per unit of variance). Large $\lambda$ penalises timing risk heavily, favouring fast execution; small $\lambda$ accepts more risk in return for lower market impact costs.

Expected shortfall:

$\mathbb{E}[\mathrm{IS}] = \gamma \sum_{k=1}^N n_k \sum_{j=1}^k n_j + \frac{\eta}{\tau} \sum_{k=1}^N n_k^2 = \frac{\gamma}{2}X^2 + \frac{\eta}{\tau}\sum_{k=1}^N n_k^2,$

using the identity $\sum_k n_k \sum_{j=1}^k n_j = \frac{1}{2}\bigl[(\sum_k n_k)^2 + \sum_k n_k^2\bigr] - \frac{1}{2}\sum_k n_k^2$ and noting $\sum_k n_k = X$. The first term is a constant (independent of schedule), so minimising expected shortfall alone reduces to minimising $\sum_k n_k^2$, which favours spreading trades as evenly as possible (TWAP).

Variance of shortfall:

$\mathrm{Var}[\mathrm{IS}] = \sigma^2 \tau \sum_{k=1}^N X_k^2,$

where $X_k = X - \sum_{j=1}^k n_j$ is the remaining inventory after period $k$. This follows because the timing risk is $\sigma\sqrt{\tau}\sum_k n_k \sum_{j=1}^k\xi_j = \sigma\sqrt{\tau}\sum_j \xi_j X_j$, and the $\xi_j$ are independent.

Continuous-Time Formulation

In continuous time with selling rate $v(t)$ (shares per unit time):

$\mathbb{E}[\mathrm{IS}] = \frac{\gamma}{2}X^2 + \eta\int_0^T v(t)^2\, dt,$

$\mathrm{Var}[\mathrm{IS}] = \sigma^2 \int_0^T x(t)^2\, dt,$

where $x(t) = X - \int_0^t v(s)\, ds$ is the remaining inventory. The objective is:

$\min_{v(\cdot)} \left\{\eta\int_0^T v(t)^2\, dt + \frac{\lambda\sigma^2}{2}\int_0^T x(t)^2\, dt\right\},$

subject to $x(0) = X$, $x(T) = 0$, $v(t) = -\dot{x}(t) \geq 0$.

Closed-Form Solution via Calculus of Variations

The Euler-Lagrange equation for the functional $\int_0^T [\eta v^2 + \frac{\lambda\sigma^2}{2}x^2]\, dt$ is:

$2\eta \ddot{x}(t) = \lambda\sigma^2 x(t),$