Optimal Execution

Garleanu-Pedersen and Almgren-Chriss models for minimizing execution costs

Large orders cost more than their headline price. Temporary impact pushes the price against you; permanent impact shifts it for all subsequent trades. Optimal execution models split a large order into a trajectory of smaller trades that minimize total cost.

API

Garleanu-Pedersen (GP)

Optimal for portfolio transitions where you have a decaying alpha signal. The model balances trading into the target quickly (to capture alpha before it decays) against the cost of trading too aggressively (market impact).

python
trajectory = hz.gp_optimal_trajectory(
    current_pos=0,            # current position (shares or contracts)
    target_pos=1000,          # desired position
    alpha_decay=0.1,          # alpha half-life parameter (higher = faster decay)
    temp_impact=1e-4,         # temporary impact coefficient
    perm_impact=1e-5,         # permanent impact coefficient
    n_steps=20,               # number of time steps to execute over
)
# Returns: list of floats -- position at each step

# How much to trade each step:
for i in range(1, len(trajectory)):
    trade = trajectory[i] - trajectory[i - 1]
    print(f"Step {i}: trade {trade:.0f} shares")

Almgren-Chriss (AC)

Optimal for liquidating a position with no alpha signal. The tradeoff is between market impact (trading too fast) and price risk (holding the position too long while the market moves against you).

python
trajectory = hz.ac_liquidator(
    initial_pos=10000,        # position to liquidate
    risk_aversion=1e-6,       # higher = more urgent liquidation
    temp_impact=1e-4,         # temporary impact coefficient
    perm_impact=1e-5,         # permanent impact coefficient
    n_steps=50,               # number of time steps
)
# Returns: list of floats -- remaining position at each step

GP vs AC: which to use

ScenarioModelWhy
Entering a new position based on a signalGPYour alpha decays — trade into it before the edge disappears
Liquidating a position (stop loss, risk reduction)ACNo alpha signal, pure cost minimization under uncertainty
Rebalancing a portfolioGPThe “alpha” is the distance from optimal weights, which decays as markets move
Unwinding after a strategy is turned offACNo remaining edge, just exit at minimum cost

Calibrating impact parameters

The impact coefficients temp_impact and perm_impact are market-specific. Rough calibration from microstructure invariance:

python
activity = hz.trading_activity(daily_volume, daily_vol, daily_trades)
natural_bet = hz.invariant_bet_size(activity)

# Temporary impact: price moves ~sqrt(shares/natural_bet) * daily_vol
temp_impact = daily_vol / (natural_bet ** 0.5)

# Permanent impact: typically 10-30% of temporary impact
perm_impact = 0.2 * temp_impact

Example: large prediction market order

python
# Buy 5000 contracts where daily volume is ~20000 (25% of daily volume)
trajectory = hz.gp_optimal_trajectory(
    current_pos=0,
    target_pos=5000,
    alpha_decay=0.05,       # edge decays slowly (event is days away)
    temp_impact=0.002,      # each 1000-lot moves price ~6 cents
    perm_impact=0.0005,
    n_steps=10,
)
# Front-loaded because alpha is decaying, but not VWAP-flat

When to use

  • Large orders: any order >5% of daily volume benefits from trajectory optimization.
  • Alpha strategies: GP captures decaying edge without paying excessive impact.
  • Risk reduction: AC provides the cost-optimal schedule for unwinding under uncertainty.
  • Prediction markets: thin order books make impact significant even for moderate sizes.

For small orders relative to daily volume (under 1%), impact is negligible and you can submit the full order at once.

Next

Microstructure Invariance Calibrate impact parameters from market data. Volatility Signature Measure true volatility for risk estimation.