Posterior Odds After a Trade Sequence in the Glosten-Milgrom Model
Consider a binary Glosten-Milgrom model where an asset's true value is either $V_H$ (high) or $V_L$ (low). Each period, with probability $\mu$ an informed trader arrives who knows the true value and trades accordingly (buys if $V = V_H$, sells if $V = V_L$). With probability
You observe a sequence of $k$ consecutive buys followed by $j$ consecutive sells.
- Starting from prior odds $\frac{\delta_0}{1 - \delta_0}$ (where $\delta_0 = P(V = V_H)$ is the prior probability of the high value), derive the closed-form posterior odds $\frac{\delta_{k+j}}{1 - \delta_{k+j}}$ after the entire sequence.
- Show how the posterior depends on the net order imbalance $k - j$ and interpret the result.
- Does the order in which the buys and sells arrive matter for the final posterior? Why or why not?
Hints
- Write down the probability of a buy (and a sell) conditional on each true value state. Remember that noise traders are equally likely to buy or sell.
- Compute the likelihood ratio for a single trade, then use the fact that trades are conditionally independent to chain the ratios together via multiplication.
- Notice that the likelihood ratio for a sell is the reciprocal of that for a buy. What does this imply about how the $k$ buy-factors and $j$ sell-factors combine?
Worked Solution
How to Think About It: This is a classic Bayesian updating problem in the Glosten-Milgrom framework. The key observation is that each trade is an independent signal about the asset's value, conditional on the true state. An informed trader always buys when $V = V_H$ and always sells when $V = V_L$. A noise trader buys or sells with equal probability regardless of value. So each trade is like observing a noisy binary signal, and we just need to chain together the likelihood ratios.
Quick Estimate: Suppose $\mu = 0.3$ (so $q = 0.7$), prior $\delta_0 = 0.5$, and we see $k = 5$ buys then $j = 2$ sells. Each buy has likelihood ratio $\frac{P(\text{buy} | V_H)}{P(\text{buy} | V_L)} = \frac{\mu + q/2}{q/2} = \frac{0.3 + 0.35}{0.35} = \frac{0.65}{0.35} \approx 1.857$. Each sell has likelihood ratio $\frac{P(\text{sell} | V_H)}{P(\text{sell} | V_L)} = \frac{q/2}{\mu + q/2} = \frac{0.35}{0.65} \approx 0.538$. After 5 buys and 2 sells, the posterior odds are
Formal Derivation:
In the GM model, the probability of observing a buy or sell given the true value is:
$P(\text{buy} | V_H) = \mu + \frac{q}{2} = \frac{1 + \mu}{2}, \quad P(\text{buy} | V_L) = \frac{q}{2} = \frac{1 - \mu}{2}$
$P(\text{sell} | V_H) = \frac{q}{2} = \frac{1 - \mu}{2}, \quad P(\text{sell} | V_L) = \mu + \frac{q}{2} = \frac{1 + \mu}{2}$
where we used $q = 1 - \mu$, so $\mu + q/2 = \mu + (1-\mu)/2 = (1+\mu)/2$ and $q/2 = (1-\mu)/2$.
Define the likelihood ratio (LR) for a single buy:
$\Lambda_{\text{buy}} = \frac{P(\text{buy} | V_H)}{P(\text{buy} | V_L)} = \frac{1 + \mu}{1 - \mu}$
and for a single sell:
$\Lambda_{\text{sell}} = \frac{P(\text{sell} | V_H)}{P(\text{sell} | V_L)} = \frac{1 - \mu}{1 + \mu} = \Lambda_{\text{buy}}^{-1}$
Since trades are conditionally independent given the true value, the likelihood ratio for $k$ buys followed by $j$ sells is simply the product:
$\Lambda_{k,j} = \Lambda_{\text{buy}}^k \cdot \Lambda_{\text{sell}}^j = \left(\frac{1+\mu}{1-\mu}\right)^k \cdot \left(\frac{1-\mu}{1+\mu}\right)^j = \left(\frac{1+\mu}{1-\mu}\right)^{k-j}$
By Bayes' rule, the posterior odds are the prior odds times the likelihood ratio:
$\frac{\delta_{k+j}}{1 - \delta_{k+j}} = \frac{\delta_0}{1 - \delta_0} \cdot \left(\frac{1+\mu}{1-\mu}\right)^{k-j}$
Interpretation of the Net Imbalance:
The posterior depends on the trade sequence only through the net order imbalance $k - j$, not on the total number of trades $k + j$ individually, and not on the order of arrival. This is because each buy contributes a factor of $\frac{1+\mu}{1-\mu}$ and each sell contributes $\frac{1-\mu}{1+\mu}$. A buy and a sell cancel each other out exactly in the likelihood ratio. So 10 buys and 7 sells produce the same posterior as 3 buys and 0 sells.
Order Independence:
The order does not matter. Since trades are conditionally independent given $V$, the joint likelihood factors into a product that depends only on the count of buys and sells, not their arrangement. Mathematically, multiplication is commutative, so rearranging the $\Lambda_{\text{buy}}$ and $\Lambda_{\text{sell}}$ factors does not change the product. This is a general property of Bayesian updating with exchangeable signals.
Answer:
The closed-form posterior odds after $k$ buys and $j$ sells are:
$\frac{\delta_{k+j}}{1 - \delta_{k+j}} = \frac{\delta_0}{1 - \delta_0} \cdot \left(\frac{1+\mu}{1-\mu}\right)^{k-j}$
The posterior depends only on the net order imbalance $k - j$, and the arrival order is irrelevant. Each net buy multiplies the odds by $\frac{1+\mu}{1-\mu} > 1$, pushing belief toward $V_H$, while each net sell divides by the same factor.
Intuition
This result captures a deep feature of Bayesian learning from order flow: information aggregates through the net imbalance, not the raw volume. A market maker who sees 100 buys and 97 sells should update beliefs exactly the same as if she had seen 3 buys and 0 sells. The total volume tells you about the overall activity level (and hence the precision of your inference about whether traders are informed or noise), but conditional on the GM model's parameters, only the net signed flow carries directional information. This is why real-world market makers focus heavily on net order imbalance as a key state variable.
The order-independence result is also practically important but easy to over-apply. It holds in the basic GM model because signals are exchangeable -- every trade has the same information content regardless of when it arrives. In richer models with time-varying informed trading intensity, strategic traders, or quote adjustment between trades, the order can matter a great deal. The basic GM result is the clean benchmark that helps you see when and why departures from order-independence arise.