Almgren-Chriss Framework for Optimal Portfolio Execution

Scenario

An algorithmic execution desk must liquidate a portfolio of $X$ shares over time window $[0, T]$ with $x(0) = X$ and $x(T) = 0$. Price impact consists of:

• Permanent impact parameter $\gamma$ affecting the fundamental price drift.
• Temporary impact parameter $\eta$ penalizing instantaneous trading velocity $v(t) = -\dot{x}(t)$ with linear slippage $\eta v(t)$.
• Market volatility $\sigma$ and risk aversion parameter $\lambda$.

Your Task

1. Formulate the expected liquidation shortfall $\mathbb{E}[x]$ and execution variance $\mathbb{V}[x]$.
2. Set up the calculus of variations functional $J[x] = \mathbb{E}[x] + \lambda \mathbb{V}[x]$ and derive the Euler-Lagrange second-order ODE $\ddot{x} - \kappa^2 x = 0$.
3. State the optimal hyperbolic trajectory $x(t) = X \frac{\sinh(\kappa(T-t))}{\sinh(\kappa T)}$ where $\kappa = \sqrt{\frac{\lambda \sigma^2}{\eta}}$.
4. For $X = 100,000$ shares, $T = 1.0$ day, $\sigma = 0.30$, $\eta = 2.5 \times 10^{-6}$, and $\lambda = 1.0 \times 10^{-5}$, compute $\kappa$ and the optimal trade size during the first hour of trading.

Constraints

• Boundary conditions $x(0) = X, x(T) = 0$.
• Continuous trading rate $\dot{x}(t)$.

Examples As risk aversion $\lambda \to 0$, $\kappa \to 0$, and the trajectory converges to the linear TWAP (Time-Weighted Average Price) schedule: $x(t) = X(1 - t/T)$.