Summary of key results · M05 Intraday and execution

Result	Formula
Spread decomposition	$q(p - m_t) = q(p - m_{t+\Delta}) + q(m_{t+\Delta} - m_t)$ (5.1)
Roll model	$p_t = m_t + \tfrac s2 q_t$ ; $\operatorname{Cov}(\Delta p_t, \Delta p_{t-1}) = -s^2/4$ ; $\hat s = 2\sqrt{-\hat\gamma_1}$ (5.2)–(5.4)
Roll with persistence $\rho$	$\operatorname{Cov} = -\tfrac{s^2}{4}(1-\rho)^2$ (Problem 5.2)
Glosten–Milgrom	$a = E[V\mid\text{buy}]$ , $b = E[V\mid\text{sell}]$ ; at $\theta = \tfrac12$ , $a - b = \mu\Delta V$ (5.5)–(5.6)
Kyle	$\lambda = \sigma_v/(2\sigma_u)$ , $\beta = \sigma_u/\sigma_v$ ; insider profit $= $ noise loss $= \sigma_v\sigma_u/2$ ; $\operatorname{Var}(v\mid y) = \sigma_v^2/2$ (5.7)
Square-root law	$I/P \approx Y\sigma_D\sqrt{Q/V_D}$ , $Y = O(1)$ (5.13)
VWAP	$\sum p_iv_i/\sum v_i$ ; volume-proportional schedule attains it (5.14), Prop. 5.4
Implementation shortfall	delay + execution + opportunity + fees (5.15)
AC cost	$E = \tfrac12\gamma X^2 + \varepsilon\sum\lvert n\rvert + \tfrac{\tilde\eta}{\tau}\sum n^2$ , $\tilde\eta = \eta - \tfrac12\gamma\tau$ ; $\operatorname{Var} = \sigma^2\tau\sum x^2$ (5.19)–(5.20)
AC trajectory	$x_j = X\sinh(\kappa(T - t_j))/\sinh(\kappa T)$ , $\tfrac{2}{\tau^2}(\cosh\kappa\tau - 1) = \lambda\sigma^2/\tilde\eta$ ; continuous $\kappa = \sqrt{\lambda\sigma^2/\eta}$ (5.21), (5.23)
Sign strategy	$E[\operatorname{sgn}(x)y] = \rho\sigma_y\sqrt{2/\pi}$ (5.24)
Cost drag	$\text{SR}_A = \sqrt N(e - c)/\sigma_t$ ; drag $\sqrt N c/\sigma_t$ ; $\text{SR}_A(h) = \sqrt H(a - c/h)/\sigma_D$ (5.25)–(5.26)
Time to t = 2	$T_2 = 4/(N s_t^2)$ years