Almgren-Chriss optimal execution, VWAP and TWAP

"""Exercise 5.3: Almgren-Chriss optimal execution, VWAP and TWAP (notes.md §5.6-5.7).

Sell X shares over [0, T] in N equal intervals of length tau = T / N.
Holdings x_0 = X, x_1, ..., x_N = 0; the k-th child order sells n_k = x_{k-1} - x_k.

Price dynamics (linear impact):
    S_k = S_{k-1} + sigma sqrt(tau) xi_k - gamma n_k          (permanent impact)
    execution price of child k = S_{k-1} - epsilon sgn(n_k) - (eta / tau) n_k   (temporary impact)

Implementation shortfall IS = X S_0 - sum n_k * (execution price) has
    E[IS] = gamma X^2 / 2 + epsilon sum |n_k| + (eta_tilde / tau) sum n_k^2,
    Var[IS] = sigma^2 tau sum_{k=1}^{N} x_k^2,
    eta_tilde = eta - gamma tau / 2          (notes eq. (5.20)).

Proposition 5.5: minimising E + lam Var gives
    x_j = X sinh(kappa (T - t_j)) / sinh(kappa T),
    (2 / tau^2)(cosh(kappa tau) - 1) = lam sigma^2 / eta_tilde.

Check:
    cd exercises && .venv/bin/python -m pytest modules/M05-intraday-execution/tests
"""

from __future__ import annotations

import numpy as np  # noqa: F401
from scipy.optimize import minimize  # noqa: F401


def eta_tilde(eta: float, gamma: float, tau: float) -> float:
    """eta - gamma * tau / 2."""
    raise NotImplementedError


def ac_cost_mean_var(x: np.ndarray, T: float, sigma: float, eta: float, gamma: float, epsilon: float = 0.0) -> tuple[float, float]:
    """(E[IS], Var[IS]) for holdings x (length N + 1, x[0] = X, x[-1] = 0) using the formulas above."""
    raise NotImplementedError


def ac_kappa(T: float, N: int, sigma: float, eta: float, gamma: float, lam: float) -> float:
    """Solve (2 / tau^2)(cosh(kappa tau) - 1) = lam sigma^2 / eta_tilde for kappa > 0 (use arccosh)."""
    raise NotImplementedError


def ac_trajectory(X: float, T: float, N: int, sigma: float, eta: float, gamma: float, lam: float) -> np.ndarray:
    """Closed-form optimal holdings x_0..x_N (length N + 1).

    x_j = X sinh(kappa (T - t_j)) / sinh(kappa T), t_j = j T / N.
    For lam == 0 return the straight line X (1 - t_j / T).
    """
    raise NotImplementedError


def ac_numerical(X: float, T: float, N: int, sigma: float, eta: float, gamma: float, lam: float) -> np.ndarray:
    """Minimise E + lam Var numerically over x_1..x_{N-1} with scipy.optimize.minimize.

    Return the full holdings vector x_0..x_N. Hints: the gamma X^2/2 and
    epsilon X terms do not depend on the schedule for a pure sell program;
    rescale to w = x / X and divide the objective by eta_tilde X^2 / tau so the
    problem is well conditioned; supply the analytic gradient; use BFGS with a
    tight gtol (1e-12).
    """
    raise NotImplementedError


def ac_continuous(X: float, T: float, t: np.ndarray, sigma: float, eta: float, lam: float) -> np.ndarray:
    """Continuous-time limit x(t) = X sinh(k (T - t)) / sinh(k T), k = sqrt(lam sigma^2 / eta)."""
    raise NotImplementedError


def twap_schedule(X: float, N: int) -> np.ndarray:
    """N child orders of X / N each."""
    raise NotImplementedError


def vwap(prices: np.ndarray, volumes: np.ndarray) -> float:
    """sum p_i v_i / sum v_i."""
    raise NotImplementedError


def vwap_schedule(X: float, volume_profile: np.ndarray) -> np.ndarray:
    """Child orders proportional to an expected volume profile: X v_k / sum v."""
    raise NotImplementedError


def implementation_shortfall_bps(side: str, decision_price: float, fill_prices: np.ndarray, fill_qty: np.ndarray) -> float:
    """Shortfall of the executed quantity versus the decision price, in bps; positive = cost.

    buy:  (avg_fill - decision) / decision * 1e4
    sell: (decision - avg_fill) / decision * 1e4
    with avg_fill the VWAP of the fills.
    """
    raise NotImplementedError