Quantitative Finance
Almgren-Chriss Optimal Execution Market Impact & Urgency Kappa Formula
Calculate expected implementation shortfall E[Cost] = 0.5 * gamma * X^2 + (eta / tau) * SUMPRODUCT(n_k, n_k) and execution urgency kappa = SQRT(lambda_risk * sigma^2 / eta) across child-order slices.
Balance temporary child-order slippage against timing risk!
In the Almgren & Chriss (2001) execution framework, liquidating X total shares across child-order slices n_1, ..., n_N over slice interval tau incurs two distinct market-impact costs: Permanent Impact = 0.5 * gamma * X^2 (which depends only on total parent size X, regardless of slicing) plus Temporary Impact = (eta / tau) * Sum(n_k^2) (which penalizes front-loaded child slices quadratically via SUMPRODUCT(n_range, n_range))!
Traders choose how aggressively to front-load slices using the Almgren-Chriss urgency parameter kappa = SQRT(lambda_risk * sigma^2 / eta): higher volatility sigma or risk aversion lambda_risk increases kappa (faster front-loaded execution), whereas higher temporary impact eta lowers kappa toward equal-slice TWAP!
| Schedule (X = 100,000 sh, 4 Days) | Permanent Cost ($) | Temporary Cost ($) | Expected Shortfall ($) |
|---|---|---|---|
| Front-Loaded [40k, 30k, 20k, 10k] | $1,000.00 | $6,000.00 | $7,000.00 |
| Equal-Slice TWAP [25k, 25k, 25k, 25k] | $1,000.00 | $5,000.00 | $6,000.00 |
| Urgency Parameter kappa (1/day) | sigma = $0.80 | lambda = 1e-5, eta = 2e-6 | 1.7889 / day |
Expected Total Market-Impact Cost ($ Implementation Shortfall)
Sums permanent impact 0.5 * gamma * X^2 and temporary slice impact (eta / tau) * SUMPRODUCT(B6:B9, B6:B9), verifying child slices sum to parent order X.
=IF(OR(B2 <= 0, B3 < 0, B4 <= 0, B5 <= 0, ABS(SUM(B6:B9) - B2) > 0.01), "Invalid Execution Inputs", ROUND(0.5 * B3 * (B2 ^ 2) + (B4 / B5) * SUMPRODUCT(B6:B9, B6:B9), 2))Almgren-Chriss Execution Urgency Parameter (kappa)
Computes continuous-time trajectory decay rate SQRT(lambda_risk * sigma^2 / eta).
=IF(OR(B4 <= 0, B10 < 0, B11 < 0), "Invalid Risk Inputs", ROUND(SQRT(B11 * (B10 ^ 2) / B4), 4))Try the example
Use the original values or change them to test your understanding.