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.

(a) 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?

(b) Do the arbitrageurs have gains or suffer losses overall from period 1 to period 3 if the noise trader sentiment deepens in period 2?