Let c1(S, t) and c2(S, t) be the prices at time t of two European call options on the same non-dividend paying stock wit

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Let c1(S, t) and c2(S, t) be the prices at time t of two European call options on the
same non-dividend paying stock with price S, with same expiration T and with strike
prices K1 and K2, respectively. Assume that K1 < K2.
(i) Explain why c1 − c2 is a solution of the Black-Scholes PDE.
(ii) By considering c1(ST , T) − c2(ST , T) deduce that
0 ≤ c1(St
, t) − c2(St
, t) ≤ (K2 − K1)e
−r(T −t)
(Hint: You need to construct arbitrage argument above in the left and right hand
sides of the inequalities.)
Let G(S,t) and cr(s, t) be the prices at time t of two European call options on the same non-dividend paying stock with price S, with same expiration T and with strike prices K, and K2, respectively. Assume that Ki < Ky. (i) Explain why 6 -c is a solution of the Black-Scholes PDE. (ii) By considering (Sr, T) - (Sr, T') deduce that OS (S,t) - (St) (K2 - Kier(T-) (Hint: You need to construct arbitrage argument above in the left and right hand sides of the inequalities.)