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sols15_28

# sols15_28 - Mich 2010 Mathematical Finance Short Solutions...

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Mich 2010 Mathematical Finance: Short Solutions 3 15. Apply the Law of One Price to some suitable portfolios. The inequalities for the put option are ( Ke - rT - S ) + P Ke - rT . For the final inequality compare portfolios containing put options and some cash. 16. Adapt the argument for the American call option i.e. compare exercising the call at T 1 with selling short. 17. Buy the calls with strike prices K 1 and K 3 and sell two calls with strike K 2 . Whatever the stock price S ( T ) the return is non-negative so the portfolio requires a positive amount to set-up. Buying the K 1 and K 3 puts and selling two K 2 puts also guarantees a positive return so P 2 ( P 1 + P 3 ) / 2. 18. Start with put-call parity for both parts: (a) is not true (think about large K and small S ), (b) is true. 19. Modify the argument used to establish Theorem 1.3 (b). 20. The initial odds have (1 + o i ) - 1 = 1 so there is no arbitrage. With natural labels for the three new bets we find o 12 = 1 / 5, o 13 = 1 / 2 and o 23 = 1.
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