We consider a single-period model with three securities: the bank account whose price process is A(0) = A(1) = 1, and two stocks with price processes given by S1(0) = s for some s > 0,

S1(1) = 1.3 in scenario 1, 0.3 in scenario 2, 0.3 in scenario 3

and S2(0) = 1.1,

S2(1) = 1.6 in scenario 1, 1.1 in scenario 2, 0.6 in scenario 3

(a) Find all risk neutral probabilities depending on s.

(b) Consider a model consisting only of the bank account and the first stock. Determine all risk-neutral probabilities (depending on the parameter s).

(c) Consider a model consisting only of the bank account and the second stock. Determine all risk-neutral probabilities.

(d) Let s = 0.9. Find an arbitrage opportunity for the model consisting of the three securities.

(e) In (d), is there an arbitrage opportunity if transaction costs of 10% apply on the transaction volume of the first stock (no transaction costs on the second stock and the bank account)