Almgren-Chriss Optimal Execution & Market Impact Lab

1. The Mean-Variance Trade-off in Portfolio Execution

Institutional asset managers executing multi-million dollar equity blocks face a fundamental conflict between market impact and market volatility risk. Liquidating an entire position immediately drives prices aggressively against the trader (temporary market impact). Conversely, spreading the trade over multiple days to minimize impact leaves the portfolio exposed to random walk market volatility and adverse price drift (variance of capture).

Robert Almgren and Neil Chriss (2000) formalized this problem by constructing a utility function $U(x) = E[x] + \lambda V[x]$, where $E[x]$ is expected implementation shortfall, $V[x]$ is execution variance, and $\lambda$ is trader risk aversion.

2. Permanent vs. Temporary Price Impact

Almgren and Chriss split market microstructure impact into two distinct physical mechanisms:

• Permanent Price Impact $g(v_k) = \gamma v_k$: Reflects the permanent information content leaked to the market. Each share sold permanently depresses the equilibrium stock price by $\tau \gamma v_k$. Across the entire order $X_0$, total expected permanent impact cost is: $$E[\text{Permanent}] = \frac{1}{2} \gamma X_0^2$$ Crucially, permanent impact is path-independent: it depends solely on total shares $X_0$ and is completely invariant to execution speed or schedule!
• Temporary Price Impact $h(v_k) = \epsilon \text{sgn}(v_k) + \eta v_k$: Reflects the instantaneous liquidity surcharge paid for demanding immediacy, crossing the bid-ask spread ($\epsilon$), and walking down limit order books ($\eta v_k$). This cost dissipates immediately after each trade. Total temporary cost is: $$E[\text{Temporary}] = \epsilon X_0 + \eta \sum_{k=1}^N \frac{n_k^2}{\tau}$$ Temporary impact is strictly path-dependent and convex: rapid trading speeds incur quadratic penalties.

3. Variance of Capture (Execution Risk)

Assuming the unaffected asset price follows an arithmetic Brownian motion $dS = \sigma dW$, the total variance of the captured cash flow is: $$V[x] = \sigma^2 \sum_{k=1}^N \tau x_k^2$$ Holding unhedged inventory $x_k$ across time slices exposes the capital to cumulative volatility $\sigma^2 \tau$.

4. Optimal Hyperbolic Trajectory Solution

Applying the calculus of variations (Euler-Lagrange equation) to minimize $U(x) = E[x] + \lambda V[x]$ yields the harmonic oscillator differential equation $\ddot{x}(t) = \kappa^2 x(t)$, with optimal urgency parameter: $$\kappa = \sqrt{\frac{\lambda \sigma^2}{\eta}}$$ Under boundary conditions $x(0) = X_0$ and $x(T) = 0$, the exact optimal trajectory is given by the ratio of hyperbolic sine functions: $$x(t) = \frac{\sinh(\kappa(T - t))}{\sinh(\kappa T)} X_0$$ The position half-life (time required to liquidate 50% of starting shares) is $t_{1/2} = \frac{\ln(2)}{\kappa}$.