{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

6 Options II

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online