In the example, the value of the asset is set at V = 1. The size of the fund managed by the

arbitrageur in period 1 is F1 = 0.2. The size of the fund in period 2 is F2 = F1

G(x), where x

is the gross return of the fund from period 1 to period 2 and G(x) = ax +1 - a with a = 1.2.

The pessimism of the niose traders in period 1 is S1 = 0.3. The pessimism of the noise traders

in period 2 is S2 = 0.4 when the pessimism deepens. It turns out that for these parameters there

is a q* = 0.35 such if the probability that the pessimism of the noise traders widens in period 2,

q, is less than q*, the arbitrageur would not hold cash in period 1. If q is greater than q, the

arbitrageur would hold some cash in period 1.

1. If q < 0.35, we have the following in the model:

* Arbitrageurs are fully invested and D1 = F1 = 0.2; the first period price is p1 = 0.9;

* F2 = 0.1636 and p2 = 0.7636 if noise trader sentiment deepens;

* F2 = 0.227 and p2 =V = 1 if noise trader sentiment recovers.

Please answer the following questions.

(a) What are the positions of the fund portfolio constructed in period 1? What

is the value of this portfolio in period 2 when the noise trader sentiment deepens?

What are the gains or losses due to the price movements?

(b) When the noise trader sentiment deepens in period 2, is the value of the

portfolio constructed in period 1 the same as the size of the fund in period 2? Explain.

(c) What are the positions of the fund portfolio constructed in period 2 when

the noise trader sentiment deepens? What are the adjustments of the portfolio made

by the arbitrageurs in period 2 in this case? What are the motivations for these

adjustments? Are these motivations consistent with the market conditions?

(d) What is the value in period 3 of the portfolio constructed in period 2

when the noise trader sentiment deepens? What are the gains or losses due to the

price movements?

(e) (1 points) Do the arbitrageurs have gains or suffer losses overall from period 1 to

period 3 if the noise trader sentiment deepens in period 2?

2. If q = 0.5, we have the following in the model:

* D1 = 0.1743 and p1 = 0.8743;

* If noise trader pessimism deepens in period 2, then F2 = 0.1766 and p2 = 0.7766;

* If noise trader sentiment recovers in period 2, then F2 = 0.23 and price returns to

V = 1.

(a) (3 points) Comparing this case (q = 0.5) with the previous case q < 0.35, is the asset

price in period 1 more or less efficient? Explain the reason.