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The **effective spread** matters most for execution quality.  The difference
between effective and realized spread measures adverse selection — what the
market maker loses to informed flow.

---

price_change = lambda * net_order_flow

import numpy as np
from numpy.linalg import lstsq

def estimate_kyle_lambda(
    price_changes: np.ndarray,
    signed_volumes: np.ndarray,
) -> float:
    """Estimate Kyle's lambda from trade data.

    Args:
        price_changes: Midprice changes between trades.
        signed_volumes: Trade volume * trade_sign (+1 buy, -1 sell).

    Returns:
        Estimated lambda (price impact per unit volume).
    """
    X = signed_volumes.reshape(-1, 1)
    beta, _, _, _ = lstsq(X, price_changes, rcond=None)
    return float(beta[0])

total_impact = permanent_impact + temporary_impact
permanent = gamma * (shares / ADV)
temporary = eta * (shares / time_horizon) ^ alpha

def square_root_impact(
    order_size: float,
    daily_volume: float,
    volatility: float,
    impact_coefficient: float = 0.1,
) -> float:
    """Estimate price impact using the square root model.

    Args:
        order_size: Number of units to trade.
        daily_volume: Average daily volume.
        volatility: Daily return volatility (decimal).
        impact_coefficient: Empirical constant (typically 0.05-0.20).

    Returns:
        Expected price impact as a fraction.
    """
    return impact_coefficient * volatility * (order_size / daily_volume) ** 0.5

def order_book_imbalance(
    bid_qty: float,
    ask_qty: float,
) -> float:
    """Compute order book imbalance.

    Returns:
        Imbalance in [-1, 1]. Positive = more bids (bullish).
    """
    total = bid_qty + ask_qty
    if total == 0:
        return 0.0
    return (bid_qty - ask_qty) / total

intensity(t) = mu + sum(alpha * exp(-beta * (t - t_i)))

reservation_price = midprice - q * gamma * sigma^2 * T
optimal_spread = gamma * sigma^2 * T + (2/gamma) * ln(1 + gamma/k)

def vwap(prices: list[float], volumes: list[float]) -> float:
    """Compute VWAP from trade prices and volumes."""
    pv_sum = sum(p * v for p, v in zip(prices, volumes))
    v_sum = sum(volumes)
    return pv_sum / v_sum if v_sum > 0 else 0.0

undefined

implementation_shortfall = (execution_price - decision_price) * quantity

def maker_pnl_per_trade(
    spread_captured_bps: float,
    maker_fee_bps: float,
    adverse_selection_bps: float,
) -> float:
    """Compute market maker P&L per round trip.

    Args:
        spread_captured_bps: Half-spread captured on each side.
        maker_fee_bps: Maker fee (negative = rebate).
        adverse_selection_bps: Expected loss to informed flow.

    Returns:
        Net P&L in basis points per round trip.
    """
    gross = 2 * spread_captured_bps  # earn half-spread on each leg
    fees = 2 * maker_fee_bps         # pay/receive fee on each leg
    return gross - fees - adverse_selection_bps

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