Unformatted text preview: ⇒ p ≥ max {Ke−rT − (S0 – D0), 0} 3. Additional Arbitrage Opportunities when Bounds are Violated: a) Assume the following: rf = 10%/yr, T = 6 months, and S0 = $25, D0 = 0 (no dividends). b) Example #1 – American or European call premium too high. 1) The upper bound is c ≤ C ≤ S0. Suppose that c = $26 and S0 = $25, so c > S0. 2) Arbitrage – write/sell call option for $26, pay $25 to buy the stock to cover it, and invest the $1 at rf = 10%. By time T, you have cash worth ($1)e0.10(1/2) = $1.05, plus: • stock worth ST if option never exercised by holder; • cash worth at least K if option exercised and you sell stock to holder for K. 3) Either way, you make π > 0; arbs rushing to write (supply) this call and buy (demand) the stock will drive c down and S0 up until c ≤ S0. c) Example #2 – American put premium too high. 1) The upper bound is P ≤ K. Suppose that P = $36 and K = $35, so P > K. 2) Arbitrage – write/sell the put option for $36 and invest it at rf = 10%. • If holder exercises at time τ, you buy his stock for $35 and end with 36e0.10τ − 35 in cash plus stock worth Sτ; • If the holder never exercises, you end up with 36e0.10(1/2) = $37.85. 3) Either way, you make π > 0; arbs rushing to write this call will drive P down until P ≤ K. Ec 174 OPTIONS II p. 7 of 18 d) Example #3 – European call premium too low. 1) The lower bound is c ≥ max {S0 − Ke−rT, 0}. Suppose that S0 = $25, K = $15, and c = $9. Then c < 25 − 15e−0.10(1/2) = $10.73. 2) Arbitrage opportunity. • At time t = 0, short sell the stock to raise $25 in cash, buy the call for $9, and invest the remainder, S0 − c = $16, at rf = 10%. • At time t = T: o If ST = $20 > K = $15, exercise the option, pay $15 to buy stock and close out short position, and pocket (S0−c)erT − K = 16.82 − 15 = $1.82/share. o If ST = $10 < K = $15, option expires. P...
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This document was uploaded on 02/18/2014 for the course ECON 174 at UCSD.
 Winter '08
 Foster,C

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