Glosten–Milgrom dynamics

$V_H = 110$, $V_L = 90$, prior $\theta = 0.3$, $\mu = 0.25$. (a) Compute the bid, ask and spread. (b) A buy arrives. Update $\theta$ and compute the new quotes. (c) Then a sell arrives. Show that $\theta$ returns exactly to its prior value 0.3, and explain why. (d) Show from (5.5) that the spread tends to 0 as $\theta \to 0$ or $\theta \to 1$ for fixed $\mu \in (0,1)$.