Question

how is this not clear?

Consider a family of 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 the following two general relations using arbitrage arguments: OV 1. If K2 > K1, then K2 – K1 > C(K1) – C(K2). 2. If K3 > K2 > K1, then C(K») (59 = K ) C(K:) + (K3 - Kj)C(Ks).

Answer 1

In both parts, one can see that there is guaranteed to have no negative but sometimes positive final payoffs. In each case below proves the following two general relations :

1. Verify that the relations - C(K₁) + C(K₂) + (K₂ - K₁) has no negative but sometimes positive final payoffs. Hence, its value must be nonnegative, thus it establishes the
result.
2. Consider the portfolio mC(K₁) - C(K₂) + nC(K₃) with m, n > 0 and m < 1. This portfolio’s final payoffs are zero if S_T ≤ K1 and will be positive on the interval K1 ≤ ST ≤ K2. Since m < 1 the payoffs decline on the interval K2 ≤ ST ≤ K3. If we set m so that

m(K₃- K₁) = (K₃ - K₂),

then the payoffs will remain nonnegative in the interval K_{​​​​​​2} ≤ ST ≤ K_3. In particular,the payoff when S_T = K₃ will be zero. If we further set n so that m + n = 1, the final payoffs will be zero on the interval S_T ≥ K₃. We conclude that the portfolio

[(K_3- K₂ )/(K₃ - K₁)] *C(K₁) - C(K₂) + [(K₂ - K₁)/
(K₃ - K₁) *C(K₃)

has no negative but sometimes positive final payoffs. Hence, its value must be nonnegative, thus it establishes the result.