Kyle (1985) and Glosten-Milgrom (1985)

"""Exercise 5.2: Kyle (1985) and Glosten-Milgrom (1985) (notes.md §5.3-5.4).

Kyle, one period. v ~ N(p0, sigma_v^2) is the liquidation value, known only
to the insider. Noise traders submit u ~ N(0, sigma_u^2), independent of v.
The insider submits x = beta (v - p0). The market maker sees only total
order flow y = x + u and sets p = p0 + lam y = E[v | y].

Proposition 5.3: lam = sigma_v / (2 sigma_u), beta = sigma_u / sigma_v,
insider expected profit sigma_v sigma_u / 2, Var(v | y) = sigma_v^2 / 2.

Glosten-Milgrom, one trade. v in {v_low, v_high}, P(v_high) = theta. A
fraction mu of arriving traders is informed (buys iff v = v_high, else
sells); the others buy or sell with probability 1/2 each. A competitive
dealer quotes ask = E[v | buy] and bid = E[v | sell] (Proposition 5.2).

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

from __future__ import annotations

import numpy as np  # noqa: F401


def kyle_equilibrium(sigma_v: float, sigma_u: float) -> dict:
    """Return {"lam", "beta", "insider_profit", "posterior_var"} from Proposition 5.3."""
    raise NotImplementedError


def simulate_kyle(p0: float, sigma_v: float, sigma_u: float, n: int, seed: int = 0, beta: float | None = None, lam: float | None = None) -> dict:
    """Simulate n independent one-period games.

    rng = np.random.default_rng(seed); draw Z1 then Z2 as two length-n standard
    normal arrays; v = p0 + sigma_v Z1; u = sigma_u Z2; x = beta (v - p0);
    y = x + u; p = p0 + lam y. beta and lam default to the equilibrium values
    when None (use them to test off-equilibrium strategies).

    Return a dict with numpy arrays "v", "u", "x", "y", "p" and
    "profit" = (v - p) * x, the insider's realised profit per game.
    """
    raise NotImplementedError


def estimate_lambda(v: np.ndarray, y: np.ndarray) -> float:
    """OLS slope, with intercept, of v on y: sum (y - ybar)(v - vbar) / sum (y - ybar)^2.

    In data, v is the price after the information has come out, so this is the
    empirical price-impact regression of later price on signed order flow.
    """
    raise NotImplementedError


def glosten_milgrom_quotes(v_high: float, v_low: float, theta: float, mu: float) -> tuple[float, float]:
    """(bid, ask) = (E[v | sell], E[v | buy]) by Bayes' rule (Proposition 5.2)."""
    raise NotImplementedError


def simulate_gm_dealer_pnl(v_high: float, v_low: float, theta: float, mu: float, n: int, seed: int = 0) -> tuple[float, float]:
    """Simulate n independent single-trade rounds at the Glosten-Milgrom quotes.

    Each round: draw v (v_high w.p. theta), draw informed (w.p. mu); an informed
    trader buys iff v = v_high; an uninformed one buys w.p. 1/2. The dealer's
    P&L is ask - v on a buy and v - bid on a sell.

    Return (mean P&L, standard error of the mean). Zero-profit quotes give a
    mean indistinguishable from 0.
    """
    raise NotImplementedError