ATR and trailing stops

"""M04A exercise 3: ATR and trailing stops (notes §4A.7-4A.8).

ATR. True range TR_t = max(H_t, C_{t-1}) - min(L_t, C_{t-1}); Wilder's ATR with
period n is W_n = mean(TR_1..TR_n), then W_t = ((n-1) W_{t-1} + TR_t)/n.
For a continuously traded log price with daily volatility sigma and no gaps,
E[H-L] ~ 2 sqrt(2/pi) sigma S (M09 Proposition 9.1), so sigma ~ ATR/(2 sqrt(2/pi) S).

Trailing stop. A long position in X_t = x0 + mu t + sigma W_t exits at the first
time the drawdown from the running maximum M_t reaches d: tau = inf{t: M_t - X_t >= d}.
Proposition 4A.11 (Taylor 1975; Lehoczky 1977): M_tau - x0 is exponential with mean
    m(d) = (exp(g d) - 1)/g,  g = 2 mu / sigma^2   (m = d when mu = 0),
so E[P&L] = m - d, P(loss) = 1 - exp(-d/m), E[tau] = (m - d)/mu (= d^2/sigma^2 if mu = 0),
and the give-back from the peak, M_tau - X_tau, equals d under continuous monitoring.
"""

from __future__ import annotations

import math

import numpy as np
import pandas as pd

RANGE_MEAN = 2.0 * math.sqrt(2.0 / math.pi)  # E[max W - min W] on [0,1], about 1.5958
BETA = 0.5825971579390106


def true_range(ohlc: pd.DataFrame) -> pd.Series:
    """TR_t from columns open/high/low/close (same index). The first bar has no
    previous close: use H - L."""
    raise NotImplementedError


def atr_wilder(ohlc: pd.DataFrame, n: int = 14) -> pd.Series:
    """Wilder's ATR: NaN for positions 0..n-2, mean(TR[0..n-1]) at position n-1,
    then ATR_t = ((n-1) ATR_{t-1} + TR_t)/n."""
    raise NotImplementedError


def atr_to_sigma(atr: float, price: float) -> float:
    """Daily log volatility implied by an ATR under the no-gap Brownian model: atr/(RANGE_MEAN*price)."""
    raise NotImplementedError


def trailing_stop_exit(path: np.ndarray, d: float) -> tuple[int, float, float]:
    """Exit of a long trailing stop of distance d on a monitored price path.

    path[0] is the entry price. Exit at the first index n >= 1 with
    max(path[:n+1]) - path[n] >= d, at price path[n]. If never triggered, exit at
    the last index. Returns (exit_index, exit_price, running_max_at_exit).
    """
    raise NotImplementedError


def trailing_analytic(mu: float, sigma: float, d: float) -> dict:
    """Closed forms of Proposition 4A.11 for continuous monitoring.

    Returns dict(mean_peak, mean_pnl, p_loss, mean_time) where mean_peak = E[M_tau - x0].
    Handle mu -> 0 continuously (use math.expm1).
    """
    raise NotImplementedError


def simulate_trailing(
    mu: float, sigma: float, d: float, dt: float, n_paths: int, seed: int, max_steps: int = 200_000
) -> dict:
    """Monte Carlo of a trailing stop monitored every dt on X = mu t + sigma W, x0 = 0.

    Use np.random.default_rng(seed). The running max starts at 0 and is updated on
    the grid; exit at the first grid time with max - X >= d.
    Returns dict(mean_peak, mean_pnl, p_loss, mean_time, mean_giveback, se_pnl), where
    peak is the observed running max at exit, giveback = peak - exit value, p_loss =
    fraction of exits below 0, se_pnl = std(pnl, ddof=1)/sqrt(n_paths).
    """
    raise NotImplementedError