ATR, Parkinson and volatility-scaled stops

(a) Show that $\mathrm{TR}_t\ge H_t-L_t$ with equality iff $L_t\le C_{t-1}\le H_t$. (b) Under the no-gap Brownian model, derive $\mathrm{ATR}\approx2\sqrt{2/\pi}\,\sigma S$ from M09 Proposition 9.1. (c) A ₹1,800 stock has ATR(14) of ₹45. Estimate its daily volatility. (d) Compute the driftless probability that a 2.5-ATR stop is hit within 15 trading days, and the share quantity for a ₹15,000 risk budget.