Almgren-Chriss Discrete Optimal Execution Dynamic Programming

The Almgren-Chriss framework is the seminal mathematical foundation for optimal portfolio execution in quantitative finance.

A quantitative fund must liquidate a parent position of $X$ shares over $T$ discrete trading intervals $t = 1, 2, \dots, T$. Let $n_t$ denote the shares sold in interval $t$, and let $q_t = X - \sum_{k=1}^t n_k$ be the remaining inventory after trading at step $t$, with boundary condition $q_T = 0$.

The objective is to minimize the total expected cost plus variance risk penalty: $\mathcal{J} = \sum_{t=1}^T \left( \eta n_t^2 + \lambda \sigma^2 q_t^2 \right)$ where:

• $\eta > 0$ represents the coefficient of temporary market impact.
• $\sigma > 0$ is the asset price volatility.
• $\lambda \ge 0$ is the risk-aversion parameter.

Your Task

Implement almgren_chriss_optimal_execution(x_total: int, t_steps: int, eta: float, sigma: float, lambda_risk: float) -> tuple[list[float], float] that calculates the exact optimal liquidation trajectory $(n_1, n_2, \dots, n_T)$ and the resulting minimum objective cost $\mathcal{J}$ using Riccati backward induction dynamic programming.

Constraints

• $1 \le x\_total \le 10^7$
• $1 \le t\_steps \le 1000$
• $\eta > 0, \sigma > 0, \lambda \ge 0$

Examples

# For zero risk aversion (lambda = 0), TWAP is strictly optimal:
trades, cost = almgren_chriss_optimal_execution(1000, 4, eta=0.01, sigma=0.1, lambda_risk=0.0)
# trades = [250.0, 250.0, 250.0, 250.0]
# cost = 4 * 0.01 * 250^2 = 2500.0