Summary of key results · M04A Reading markets

Result	Formula
Bar aggregation	$O=O_1$ , $H=\max H_i$ , $L=\min L_i$ , $C=C_k$ , $V=\sum V_i$ (4A.1)
Candle shapes under BM	law of (body, wicks)/range is timeframe-free (Prop. 4A.1); doji ≈ 9.8%, hammer shape ≈ 2.9%
Expected range	$\mathbb E[H-L]\approx1.596\,\sigma\sqrt\Delta\,S$
Touch probability	$2\Phi(-\delta/(\sigma\sqrt T))$ ; after a touch, bounce vs break is $\tfrac12$ (Prop. 4A.2)
Close minus VWAP	$\operatorname{Var}=\sigma^2(T-2\int wt+\iint ww\min)$ ; uniform volume $\sigma^2T/3$ (4A.3)
Gap fill	$2\Phi(-\lvert g\rvert/\sigma_{in})$ (4A.4)
Breadth, one factor	$\mathbb E[A/N\mid f]=\frac1N\sum\Phi(\beta_if/s_i)$ (4A.5)
Bracket, no drift	$p=b/(a+b)$ , $\mathbb E\tau=ab/\sigma^2$
Bracket, drift	$p=(1-e^{\gamma b})/(e^{-\gamma a}-e^{\gamma b})$ , $\gamma=2\mu/\sigma^2$ (4A.6); $\mathbb E\tau=(ap-b(1-p))/\mu$ (4A.7)
Exits and expectancy	$\mathbb E X_\tau=\mu\mathbb E\tau$ for any stopping time (4A.8)
Stop over horizon	$\Phi\big(\frac{-b-\mu T}{\sigma\sqrt T}\big)+e^{-2\mu b/\sigma^2}\Phi\big(\frac{-b+\mu T}{\sigma\sqrt T}\big)$ (4A.9)
Discrete monitoring	shift barrier by $0.5826\,\sigma\sqrt{\Delta t}$ (4A.10)
ATR	$\mathrm{TR}=\max(H,C_{-1})-\min(L,C_{-1})$ (4A.11); $\hat\sigma=\mathrm{ATR}/(1.596S)$ (4A.12); $k$ -ATR stop hit in $n$ days $2\Phi(-1.596k/\sqrt n)$ (4A.13)
Trailing stop	peak $\sim\mathrm{Exp}$ , mean $(e^{\gamma d}-1)/\gamma$ (4A.14); give-back $d$ ; $P(\text{loss})=1-e^{-d/m}$ ; $\mathbb E\tau=(m-d)/\mu$ or $d^2/\sigma^2$ (4A.15)
Any sizing rule	$\mathbb EG=\mu\mathbb E\int\theta dt$ , noise variance $\sigma^2\mathbb E\int\theta^2dt$ (4A.16)
Time stop	optimal at the signal horizon $h$ (Prop. 4A.13)
Widening a stop	expected P&L $\mu\mathbb E\tau$ , worse for wider stops when $\mu<0$ (Prop. 4A.14)