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1012 105 plus stock worth st if option never exercised

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Unformatted text preview: p. 5 of 18 B. Upper and Lower Bounds {Hull §10.3} Table 4 – UPPER AND LOWER BOUNDS 1. Call Options: Type Lower Upper Am. Call C ≥ max {S0 – K, 0} C ≤ S0 a) Upper bound on call premiums Am. Put P ≥ max {K – S0, 0} (both American and European). P ≤ K 1) An American call premium is Eur. Call c ≥ max {(S0 – D0) – Ke−rT, 0} c ≤ S0 always less than the stock Eur. Put p ≥ max {Ke−rT – (S0 – D0), 0} p ≤ Ke–rT price: C ≤ S0. 2) Logical argument - - Intuitively, a call is the right to buy stock at K per share. There is no reason to ever pay more for the right to buy the share than you would have to pay to buy the share itself, so C ≤ S0. (See also the arbitrage argument in Example #1 below.) 3) If European options give you less exercise opportunity than American options, we expect c ≤ C, but the upper bound result is the same: c ≤ C ⇒ c ≤ S0. b) Lower bound on American call premium. 1) An American call is always worth at least its intrinsic value, even if the time value = 0: C ≥ max {S0 – K, o}. 2) Arbitrage argument - - To see why, suppose the contrary. • If 0 < C < S0 – K, an arbitrageur pays C for the option, exercises immediately to buy stock at K and sell at S0 > K, for riskless π = S0 - K – C > 0. • The rush by arbs to buy (demand) calls and sell (supply) stock will drive C up and S0 down until C ≥ S0 – K. c) Lower bounds on a European call premium. C S0 1) c ≥ max {(S0 – D0) − Ke−rT, 0}. To see why, consider portfolios A and B below. S0 – K 2) Portfolio A = {one European call option with strike price K, plus cash worth D0 + Ke−rT} time −rT + c • Cost today is A0 = D0 + Ke • At time T, {AT} = D0erT + max {ST, K} intrinsic o If ST < K, option expires; AT = K + D0erT in K S0 cash o If ST > K, exercise option, spend K to buy Fig. 1 - - Premium stock worth ST > K; AT = ST + D0erT Bounds on Amer. Call 3) Portfolio B = {one share of stock}. • Cost today is B0 = S0 • At time T, BT = ST + D0erT 4) By Replication #2, {AT} ≥ BT, so portfolio...
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