6 Options II

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Unformatted text preview: ay \$10 to buy stock and close out short position, and pocket (S0−c)erT − ST = 16.82 − 10 = \$6.82/share. 3) Either way, you make π > 0; arbs rushing to buy (demand) this call and sell the stock will drive c up and S0 down until c ≥ S0 − Ke−rT. e) Example #4 – European put premium too low. 1) The lower bound is p ≥ max {Ke−rT − S0, 0}. Suppose that S0 = \$25, K = \$35, and p = \$8. So p < 35e−0.10(1/2) − 25 = \$8.29. 2) Arbitrage. • At time t = 0, borrow S0 + p = \$33, buy option for \$8 and buy stock for \$25. • At time t = T: o If ST = \$34 < K = \$35, exercise option, sell stock for \$35, pay 33erT = \$34.69 to close loan, and pocket 35 − 34.69 = \$0.31. o If ST = \$36 > K = \$35, option expires. Sell stock for \$36, pay off loan for \$34.69, and pocket \$1.31. 3) Either way, you make π > 0; arbs rushing to buy (demand) the put and the stock will drive p and S0 up until p ≥ Ke−rT − S0. Ec 174 OPTIONS II p. 8 of 18 C. Put- Call Parity Theorems {Hull §10.4; BKM §20.4} Table 5 - - PUT- CALL PARITY THEOREMS 1. Put- Call Parity for European Options: European options: c + Ke−rT = p + (S0 – D0) American Options: (S0 – D0) – K ≤ C – P ≤ S0 – Ke−rT a) Consider again portfolios A and L. 1) Portfolio A = {one European call, plus cash worth D0 + Ke−rT}. • Cost today is A0 = D0 + Ke−rT + c. • At time T, {AT} = D0 erT + max {ST, K}. o If ST < K, option expires; you have AT = K + D0 erT in cash o If ST > K, exercise option, spend K to buy stock worth ST > K; AT = ST + D0 erT 2) Portfolio L = {one European put (same K and T as call in A), plus one share of stock}. • Cost today is L0 = S0 + p. • At time T, {LT} = D0 erT + max {ST, K}. o If ST > K, option expires and you have stock worth ST > K; LT = ST + D0 erT o If ST < K, exercise option, sell share for K > ST; LT = K + D0 erT b)...
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