Posterior After Buy Then Sell
Consider a binary-value market-making setup: an asset has true value $V \in \{v_L, v_H\}$ with prior $P(V = v_H) = \delta$. A market maker posts a bid-ask $(b, a)$ with $b < a$. Each period, an independently drawn trader arrives who is informed with probability $\pi$ and uninformed with probability
In period 1, a buy occurs at the ask. In period 2, a sell occurs at the bid (quotes remain the same).
- Compute $P(V = v_H \mid \text{buy then sell})$.
- Compute $E[V \mid \text{buy then sell}]$.
- Are the two trades together bearish, bullish, or neutral relative to the prior?
Hints
- Write down the probability of a buy and the probability of a sell under each value of $V$, using the mixture of informed and uninformed traders.
- Compute the joint likelihood $P(\text{buy, sell} \mid V)$ for each value of $V$. Since traders are independently drawn, the joint likelihood factors.
- Compare the joint likelihoods under $v_H$ and $v_L$. What happens when you multiply $(1+\pi)/2$ by $(1-\pi)/2$?
Worked Solution
How to Think About It: This is a classic Glosten-Milgrom sequential trade model. A buy is bullish signal (informed traders buy when $V = v_H$), and a sell is bearish signal (informed traders sell when $V = v_L$). But uninformed traders generate noise in both directions. The question is: do these two opposing signals cancel out? Not quite -- the answer depends on the fraction of informed traders $\pi$. If $\pi = 0$ (all noise), the posterior equals the prior. If $\pi > 0$, the signals partially cancel but there is a subtle asymmetry to check.
Quick Estimate: By symmetry of the model, a buy followed by a sell should provide exactly offsetting information. The posterior should equal the prior $\delta$. Let us verify this formally.
Approach: Apply Bayes' theorem sequentially, computing the likelihood of observing a buy then a sell under each value of $V$.
Formal Solution:
*Step 1 -- Trade likelihoods:*
Under $V = v_H$: - $P(\text{buy} \mid v_H) = \pi \cdot 1 + (1 - \pi) \cdot \frac{1}{2} = \frac{1 + \pi}{2}$ - $P(\text{sell} \mid v_H) = \pi \cdot 0 + (1 - \pi) \cdot \frac{1}{2} = \frac{1 - \pi}{2}$
Under $V = v_L$: - $P(\text{buy} \mid v_L) = \pi \cdot 0 + (1 - \pi) \cdot \frac{1}{2} = \frac{1 - \pi}{2}$ - $P(\text{sell} \mid v_L) = \pi \cdot 1 + (1 - \pi) \cdot \frac{1}{2} = \frac{1 + \pi}{2}$
*Step 2 -- Joint likelihood of buy then sell:*
Since the two traders are drawn independently (conditional on $V$):
$P(\text{buy, sell} \mid v_H) = \frac{1 + \pi}{2} \cdot \frac{1 - \pi}{2} = \frac{1 - \pi^2}{4}$
$P(\text{buy, sell} \mid v_L) = \frac{1 - \pi}{2} \cdot \frac{1 + \pi}{2} = \frac{1 - \pi^2}{4}$
*Step 3 -- Posterior:*
By Bayes' theorem:
$P(V = v_H \mid \text{buy, sell}) = \frac{\delta \cdot \frac{1 - \pi^2}{4}}{\delta \cdot \frac{1 - \pi^2}{4} + (1 - \delta) \cdot \frac{1 - \pi^2}{4}}$
The likelihoods are identical, so they cancel:
$P(V = v_H \mid \text{buy, sell}) = \frac{\delta}{\delta + (1 - \delta)} = \delta$
*Step 4 -- Posterior mean:*
$E[V \mid \text{buy, sell}] = v_H \cdot \delta + v_L \cdot (1 - \delta) = E[V]$
This equals the prior mean.
*Step 5 -- Interpretation:*
The two trades together are neutral relative to the prior. A buy followed by a sell produces the same joint likelihood under $v_H$ and $v_L$ (both equal $\frac{1 - \pi^2}{4}$), so the likelihood ratio is
Answer: $P(V = v_H \mid \text{buy, sell}) = \delta$ (the prior). $E[V \mid \text{buy, sell}] = \delta v_H + (1 - \delta) v_L$ (the prior mean). The buy-then-sell sequence is neutral -- it conveys no net information about $V$.
Intuition
The key insight is that in this symmetric binary model, a buy and a sell provide exactly offsetting signals. The joint likelihood of observing a buy then a sell is $(1+\pi)(1-\pi)/4 = (1-\pi^2)/4$ under both $v_H$ and $v_L$. Since the likelihoods are equal, Bayes' theorem gives a likelihood ratio of
This result would not hold if the model were asymmetric -- for example, if informed traders were more likely to trade in one direction, or if the probability of being informed differed for buyers and sellers. In real markets, order flow is typically not symmetric (there is often a directional bias), which is why market makers update quotes after every trade rather than assuming buy-sell pairs cancel.