6 Options II

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Unformatted text preview: The Put- Call Parity Theorem. 1) By Replication #1, {AT} ≡ {LT}, so portfolio A must cost the same as portfolio L. 2) A0 = L0 ⇒ D0 + Ke−rT + c = S0 + p, or c + Ke−rT = p + (S0 – D0). 2. Put- Call Relationships for American Options: a) There is no put- call parity theorem for American options, but there are upper and lower bounds on the difference in premiums C – P: (S0 – D0) – K ≤ C – P ≤ S0 – Ke−rT. b) For the lower bound, consider Portfolios M and J below. 1) Portfolio M = {one European call (strike K, maturity T), plus cash = D0 +K}. • Cost today is M0 = c + D0 + K. • At time T, {MT} = (D0 + K)erT − K + max {ST, K}. o If ST < K, option expires; you have MT = (D0 + K)erT in cash o If ST > K, exercise option, spend K to buy stock worth ST; MT = (D0+K)erT − K + ST 2) Portfolio J = {one American put option with same K and T, plus one share of stock}. • Cost today is J0 = P + S0. • Case 1 - - if the option is not exercised early, then Portfolio J is the same as Portfolio L above, and {JT} = D0 erT + max {ST, K}. • Case 2 - - if the option is exercised at τ < T, then Sτ < K and you sell stock for K. o If no dividends paid by time τ, Jτ = K o If dividends paid before time τ, Jτ = D0 erτ + K 3) Portfolio M is always worth more than Portfolio J, even if D = 0. • At T, {MT} = (D0 + K)erT − K + max {ST, K} > D0 erT + max {ST, K} = {JT}. o So MT > JT for Case 1 • At τ < T, Mτ is worth at least the cash (D0 + K)erτ > D0 erτ + K > K. o So Mτ > Jτ for Case 2 4) By Replication #2, it must be the case that Portfolio M costs at least as much as J. • M0 ≥ J0 ⇒ c + D0 + K ≥ P + S0 and c – P ≥ (S0 – D0) – K. • But C ≥ c, so C − P ≥ (S0 – D0) − K, or (S0 – D0) − K ≤ C – P Ec 174 OPTIONS II p. 9 of 18 c) For the upper...
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