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ps5sol - 70-492 Investment Analysis Problem Set 5...

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70-492 Investment Analysis Problem Set 5: Derivatives 1. a) Let S T be the S&P 500 value per dollar invested, Payoff from fund = 0 . 95 if S T < 0 . 95 , S T if 0 . 95 < S T < 1 . 15 , 1 . 15 if S T > 1 . 15 , which is depicted in Figure 1. b) Payoff from call = max { S T - x, 0 } . Payoff from put = max { x - S T , 0 } . To replicate the payoff from the fun, we can invest \$1 in S&P500, purchasing a put with striking price x = 0 . 95, and sell a call with striking price x = 1 . 15. Here is how the plan works Payoff = S T + 0 . 95 - S T = 0 . 95 if S T < 0 . 95 S T if 0 . 95 < S T < 1 . 15 - ( S T - 1 . 15) + S T = 1 . 15 if S T > 1 . 15 From put-call parity, C ( x ) - P ( x ) = S T - x (1 + r ) T , which gives P ( x ) = C ( x ) - S T + x (1 + r ) T (1) The total cost of the replicating strategy is P (0 . 95) + S T - C (1 . 15) . (2) Using (1) we have C (0 . 95) - S T + 0 . 95 (1 + r ) T + S T - C (1 . 15) = C (0 . 95) - C (1 . 15) + 0 . 95 (1 + r ) T The above equation indicates that we can buy a call with a strike price of 0.95, sell a call with strike price of 1.15, and invest in bond with face value of 0.95. c) From the put-call parity, C ( x ) = P ( x ) + S T - x (1 + r ) T From (2), we have P (0 . 95) + S T - C (1 . 15) = P (0 . 95) + S T - P (1 . 15) - S T + 1 . 15 (1 + r

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ps5sol - 70-492 Investment Analysis Problem Set 5...

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