Suppose that in the Glosten$-$Milgrom model the security$$s true value v can be EUR $5$

with probability $1/5,$ EUR $3$ with probability $1/5,$ and EUR $1$ with probability $3/5.$

Market makers are competitive and risk$-$neutral, and do not know v$.$ In each period, a

single trader comes to the market: with probability $1-$ p she is a noise trader, who

buys and sells one unit with probability $(1)/(2)$ each; with probability p she is an informed

trader who knows the security$$s true value. When answering the questions under a$,$ b$,$

and c below assume that the informed trader buys only when the true value of the

security is EUR $5$ and sells only when the true value of the security is EUR $1.$

a$.$ Write down dealers$$ expected profits for the case of a buy order and the case

of a sell order.

b$.$ Compute the bid and ask prices set by risk$-$neutral competitive market makers,

and find how the ask and bid prices vary in response to an increase in the

probability p of informed trading.

c$.$ Derive the bid$-$ask spread, and find how it varies in response to an increase in

the probability p of informed trading.

d$.$ Under which condition on the parameter p does the informed trader have no

incentive to deviate from the assumed strategies of buying when the true value

of the security is EUR $5,$ selling when it is EUR $1,$ and not trading if it is EUR

$3?$