We discussed a numerical example of the professional arbitrage model in class. This question asks you how the professional arbitrage model works and what its implications are in this context.

In the example, the value of the asset is set atV=1. The size of the fund managed by the arbitrageur in period 1 isF1=0.2. The size of the fund in period 2 isF2=F1G(x), wherexis the gross return of the fund from period 1 to period 2 andG(x) =ax+1awitha=1.2. The pessimism of the niose traders in period 1 isS1=0.3. The pessimism of the noise traders in period 2 isS2=0.4 when the pessimism deepens. It turns out that for these parameters there is aq=0.35 such if the probability that the pessimism of the noise traders widens in period 2,q, is less thanq, the arbitrageur would not hold cash in period 1. Ifqis greater thanq, the arbitrageur would hold some cash in period 1.

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?