Mich 2010Mathematical Finance: Short Solutions315. Apply the Law of One Price to some suitable portfolios. The inequalities for the putoption are (Ke-rT-S)+≤P≤Ke-rT. For the final inequality compare portfolioscontaining put options and some cash.16. Adapt the argument for the American call option i.e. compare exercising the call atT1with selling short.17. Buy the calls with strike pricesK1andK3and sell two calls with strikeK2. Whateverthe stock priceS(T) the return is non-negative so the portfolio requires a positiveamount to set-up.Buying theK1andK3puts and selling twoK2puts also guarantees a positive returnsoP2≤(P1+P3)/2.18. Start with put-call parity for both parts: (a) is not true (think about largeKandsmallS), (b) is true.19. Modify the argument used to establish Theorem 1.3 (b).20. The initial odds have∑(1 +oi)-1= 1 so there is no arbitrage. With natural labelsfor the three new bets we findo12= 1/5,o13= 1/2 ando23= 1.
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