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 ω1, 0.3 in scenario ω2, 0.3 in scenario ω3
and S2(0) = 1.1,
S2(1) = 1.6 in scenario ω1, 1.1 in scenario ω2, 0.6 in scenario ω3
(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)