Answer Happy

Math finance and stochastic problem: 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 ci(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 Kį and K2, respectively. Assume that Ki < K2. > (i) Explain why Ci - C2 is a solution of the Black-Scholes PDE. . (ii) By considering cı(ST,T) – C2(ST, T) deduce that 0 <C(St,t) – cz(St, t) < (K2 – Kie-r(T—t) (Hint: You need to construct arbitrage argument above in the left and right hand sides of the inequalities.)