Roll with unbalanced and persistent order flow

Keep (5.2) and (A1), (A3). (a) Let $q_t$ be i.i.d. with $P(q_t = 1) = p$. Show that $\operatorname{Cov}(\Delta p_t, \Delta p_{t-1}) = -\tfrac{s^2}{4}\cdot 4p(1-p)$, so the Roll estimator recovers $s\sqrt{4p(1-p)}$. (This is roll_spread_theory in Exercise m05_roll.) (b) Let $q_t$ be a stationary two-state Markov chain on $\{-1,+1\}$ with $P(q_t = q_{t-1}) = (1+\rho)/2$, $\rho \in (-1, 1)$. Show that $\operatorname{Corr}(q_t, q_{t-k}) = \rho^k$, and that $\operatorname{Cov}(\Delta p_t, \Delta p_{t-1}) = -\tfrac{s^2}{4}(1-\rho)^2$ and $\operatorname{Var}(\Delta p_t) = \sigma^2 + \tfrac{s^2}{2}(1-\rho)$. (c) If true order-flow persistence is $\rho = 0.4$, by what factor does the Roll estimator understate the spread?