Question

A single period model has a bank account with interest rate r = 1/9, three states of the world, and two risky assets S1 and S2. The initial prices of S1 and S2 are 63 and 36, respectively. At time 1 the price of S1 is 80, 73, and 60 in states 1, 2, and 3, respectively. At time 1 the price of S2 is 50, 35, and 30 in states 1, 2, and 3, respectively. The three states are equally likely under the statistical probability. Are there any arbitrage opportunities? If so, specify them. If not, then specify all the risk neutral probability measures.

Answer 1

Asset S1

Expected value at time 1 = Sum[Value*probability]

Expected value at time 1 = 80*1/3+73*1/3+60*1/3 = (80+73+60)*1/3 = 71

r: interest rate = 1/9

Expected present value at time 0 = 71/(1+r) = 71/(1+1/9) = 63.9

Actual price at time 0 = 63

Since (Expected present value at time 0) > (Actual price at time 0), asset S1 is underpriced and should be bought

Asset S2

Expected value at time 1 = Sum[Value*probability]

Expected value at time 1= (50+35+30)*1/3 = 38.334

r: interest rate = 1/9

Expected present value at time 0 = 38.334/(1+r) = 38.334/(1+1/9) = 34.5

Actual price at time 0 = 36

Since (Expected present value at time 0) < (Actual price at time 0), asset S2 is overpriced and should be sold

Arbitrage opportunity

Buy asset s1 at t=0 buy borrowing money from market

Amount borrowed = 63

Buy 1 unit of asset S1

Amount to be paid at (t=1) with interest = 63*(1+1/9) = 70

Sell asset S1 at (t=1) = 71

Arbitrage profit = 71-70 = 1

Short sell asset s2 at t=0 and invest the cash

Borrow 1 unit of s2 and short-sell

Amount received = 36

Invest at 1/9 interest rate

Amount to be received at (t=1) with interest = 36*(1+1/9) = 40

Buy asset s2 from market to pay back = 38.334

Arbitrage profit = 40-38.334 = 1.666