Prove the following boundary conditions using an arbitrage argument. In your proof, show the initial positive cash flow when the condition is violated and prove there are no liabilities at expiration or when the positions are closed.

a. European futures call option: Ct ≥ Max[PV(ft − X), 0]

b. European futures put option: Pt ≥ Max[PV(X − ft), 0]

c. American spot call option: Ct ≥ Max[St − X, 0]

d. American spot put option: Pt ≥ Max[X − St, 0]

e. Put–call futures parity for European options: Pt − Ct = (X − ft)/(1+ Rf)T