Consider the sequential trading model of Glosten and Milgrom (1985). Let us assume that:

• The future value of the asset, V, can take the value of 5 or 4. The asset takes the value of 5 with probability 0.6. That is, =0.6.
• Thirty percent of traders are informed. That is, =0.3.
• Further, assume that the uninformed traders are equally likely to buy or sell.
• The dealer earns zero expected profits, such that:

at=E(V|Bt)

bt=E(V|St)

where at is the ask price, bt is the bid price, Bt indicates a buy order and St indicates a sell order all at time t.

1. Calculate the unconditional probability of observing a buy order at time t. That is, P(Bt).
2. Calculate the unconditional probability of observing a sell order at time t. That is, P(St).
3. Given Bayes theorem:

PYZ=P(Z|Y)P(Y)P(Z)

solve for the bid price at time t.

4. Solve the ask price at time t.
5. Solve for the bid-ask spread at time t.
6. Solve for the relative bid-ask spread at time t.