Consider a family of European call options on a non-dividend-paying stock, each option being identical except for its strike price. The value of the call with strike price K is denoted by C(K). Prove 1 the following two general relations using arbitrage arguments:

(a). If K₂ > K₁, then C(K₁) C(K₂). (Hint: You may argue this by contradiction. Assume that C(K₁) < C(K₂) and construct an arbitrage portfolio to obtain positive arbitrage profit. Then, that implies the conclusion must be true in an efficient market.)

(b) If K₂ > K₁, then K₂ K₁ C(K₁) C(K₂).

(c) Consider three options with strike price K₃ > K₂ > K₁, where K₃ K₂ = K₂ K₁. Show that C(K₂) (C(K₁) + C(K₃))/2.