Bayesian Update from a No-Trade Event
You are a market maker quoting a symmetric spread $[-s, +s]$ around zero for a one-shot game. The true value of the asset is $V \sim N(0, \sigma^2)$, unknown to you but observed perfectly by informed traders.
The arrival structure is as follows. With probability $p_0$, nobody arrives and no trade occurs. Conditional on arrival, with probability $\mu$ an informed trader shows up -- she buys if $V > +s$ and sells if $V < -s$, and does nothing if $|V| \leq s$. With probability
- \mu$, a noise trader arrives -- she buys or sells with equal probability $\frac{1}{2}$, independently of $V$, but declines to trade at all with probability $q$.
You observe that no trade occurs.
**(a)** Write down $P(\text{no trade} \mid V = v)$ as an explicit function of $v$, $s$, $\mu$, $p_0$, and $q$.
**(b)** Use Bayes' theorem to compute the posterior mean $E[V \mid \text{no trade}]$. Does observing no trade shift your mean estimate of $V$ away from zero? Why or why not?
**(c)** Even though the posterior mean may be unchanged, the posterior distribution is not the same as the prior. Describe qualitatively how the posterior variance $\text{Var}(V \mid \text{no trade})$ compares to $\sigma^2$, and explain what this implies for how you should requote your spread.
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