4. (20 points) Prove the following relationships. (a) (5 points) In a security market with uncertainty, suppose the interest rate r is a scalar, and let c and p denote the prices of a European call and put, respectively, both having the same strike price K. Show that the put-call parity holds, C-p=So K 1+r (b) (5 points) E(X+Y).F] = E(X\.$]+EY $). (c) (5 points) If E(A2+1 %) = 0, t = 0,1,...,1 - 1, then 2 = {20, 21, ... , Zr} is a martingale. (d) (5 points) If 2+ = E(21)F), t = 0,1,... ,T - 1, then 2 = {20,21 ,, Zr} is a martingale. 4. (20 points) Prove the following relationships. (a) (5 points) In a security market with uncertainty, suppose the interest rate r is a scalar, and let c and p denote the prices of a European call and put, respectively, both having the same strike price K. Show that the put-call parity holds, C-p=So K 1+r (b) (5 points) E(X+Y).F] = E(X\.$]+EY $). (c) (5 points) If E(A2+1 %) = 0, t = 0,1,...,1 - 1, then 2 = {20, 21, ... , Zr} is a martingale. (d) (5 points) If 2+ = E(21)F), t = 0,1,... ,T - 1, then 2 = {20,21 ,, Zr} is a martingale