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Cost today is l0 s0 p at time t lt d0 ert max

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Unformatted text preview: A must cost at least as much as B. • A0 ≥ B0 ⇒ D0 + Ke−rT + c ≥ S0 ⇒ c ≥ max {(S0 – D0) − Ke−rT, 0} 2. Put Options: a) Upper bound on an American put premium P. 1) An American put premium is always less than the strike price: P ≤ K. 2) Logical argument - - The payoff on a put is K−St per share. If St drops to zero, you get the maximum possible payoff - - $K. You would never pay more than K for an opportunity to get back K at most, so P ≤ K. Ec 174 OPTIONS II p. 6 of 18 b) Upper bound on a European put premium. 1) p ≤ Ke−rT. 2) Logical argument - - If you buy the option, you pay p today and get payoff K−ST or 0 at time T. If ST = 0, max payoff will be K, with a PV of Ke−rT. You would never pay more than Ke−rT for such an option today, so p ≤ Ke−rT. P Fig. 2 - - Premium c) Lower bounds on an American put premium. Bounds on Amer. Put 1) P ≥ K − S0. K 2) Arbitrage argument - - If P < K − S0, buy stock for S0 and the put for P, then exercise immediately and sell at K > S0 + P; instant π = K − (S0+P) > 0. time K – S0 The sudden demand for put options and stock intrinsic would drive P and S0 up until P ≥ K – S0. K S0 d) Lower bounds on a European put premium. 1) p ≥ max {Ke−rT − (S0 – D0), 0}. To see why, consider portfolios L and M below. 2) Portfolio L = {one European put option with strike price K, plus one share of stock}. • Cost today is L0 = S0 + p • At time T, {LT} = D0erT + max {ST, K} o If ST > K, option expires and you have stock worth ST > K; LT = ST + D0erT o If ST < K, exercise option, sell share for K > ST; LT = K + D0erT 3) Portfolio M = {cash worth D0 + Ke−rT}. • Cost today is M0 = D0 + Ke−rT • At time T, MT = K + D0erT 4) By replication #2, {LT} ≥ MT, so portfolio L must cost at least as much as M. • L0 ≥ M0 ⇒ S0 + p ≥ D0 + Ke−rT...
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