Consider the 3-period model that we developed in class. We assume that:

1. The final value of the security has the following probability distribution:

g(v)=12????(?|???|?)

This implies that E(v|v?z)=z+?.

2. The trader who arrive in period 2 knows the final value of the security v with probability ?, otherwise he is uninformed.

3. If the traders arrives in period 1 is uninformed, he buys or sells, with equal probability, a number of shares x that has an exponential distribution with parameter ?. That is, the size distribution of the market orders submitted by an uninformed trader arriving in period 2 is f(x)=12????|?|.

4. The tick size is nil (?=0).

a) Let Y(A) be the cumulative depth up to ask price A in the book and A? be the lowest ask price in the LOB. Show that when v?A?, the optimal strategy of the informed trader is to buy Y(v) shares. Hint: show that the optimal strategy of the informed investor is to purchase all shares up to A=v, i.e. Y(v) shares.

b) Using this observation in (a) and the zero profit condition, assuming there is no order-processing costs, show that in equilibrium

Y(A)=1?[??(1???)+??(????)+????] ?? ?>??

Hint: because the optimal strategy of the informed investor is to purchase all shares up to A=v, when the order is from an informed trader and when Y(v)>y, it is equivalent as v>A.

c) Show that the book becomes thinner on the ask side when (i) ? increases or (ii) ? increases. What is the economic intuition for this result?

Q3. Deriving a competitive LOB. Consider the 3-period model that we developed in class. We assume that: 1. The final value of the security has the following probability distribution: 1 _ s0!) = E exp ( I\" lI\") (I This implies that E(vlv 2 z) = z + o. 2. The trader who arrive in period 2 knows the final value of the security v with probability Tr, otherwise he is uninformed- 3. If the traders arrives in period 1 is uninformed, he buys or sells, with equal probability, a number of shares x that has an exponential distribution with parameter 9. That is, the size distribution of the market orders submitted by an uninformed trader arriving in period 2 is f(x) = gee9"\". 4. The tick size is nil (A = 0]. a} Let Y(A) be the cumulative depth up to ask price A in the book and A' be the lowest ask price in the LOB. Show that when v 2 A', the optimal strategy of the informed trader is to buy Y0!) shares. Hint: show that the optimal strategy of the informed investor is to purchase all shares up to A = v, i.e. Y(v) shares. b} Using this observation in (a) and the zero prot condition, assuming there is no order- processing costs, show that in equilibrium ETA)=%[in(1;)+in(A;p)-PA;M] ifA>A* Hint: because the optimal strategy of the informed investor is to purchase all shares up to A = v, when the order is from an informed trader and when Y(v) 3: y, it is equivalent as v :3- A. c} Show that the book becomes thinner on the ask side when (i) It increases or {iii o increases. What is the economic intuition for this resuit