Question 7 and 8 Please Answer c). (3 points) You w Du e the value of a three-mo

Question

Question 7 and 8 Please

Answer c). (3 points) You w Du e the value of a three-month European put option on stock with a strike price (K) of $110. 13 points) You would like to instantaneously hedge your $15,000 investment in bis stock. How many three-month at-the-money European call options you would e How many three-month at-the-money European put options would you purchase? 7. (10 points) There are points) There are three European call options with strike price 70, 75, and 80. The menonding call prices are 16, 13, and 10, respectively. Is there an arbitrage opportunity? If yes, please describe it in details. 8 (15 points) Describe how you would use a six-month E-mini S&P; 500 futures contract to perform each of woo Derfor each of the following tasks. Remember that one E-mini pays $50 times the S&P; 500 index level. (For the purposes of this question, ignore the fact that the S&P; 500 component stocks pay dividends.) The current level of the index is 1250. a). (5 points) Simulate investing $250,000 in the S&P; 500 index for six months. b). (5 points) Fully hedge an investment of $550,000 in a well diversified portfolio that has a beta of 1.2. c). (5 points) Partially hedge an investment of $550,000 in a well-diversified portfolio that has a beta of 1.2 so that the new portfolio has a beta of 0.5.

Explanation / Answer

7. There is not arbitrage in this case. In general across two strike prices of same underlying with same expiry date, arbitrage is possible when (Premium of lower strike call - Premium of higher strike call - difference in strike prices) is positive. In this case for 70 & 75 strike prices the difference in strike prices is 5 and difference in premium is only 3, and similar for 75 and 80 also - hence there is no arbitrage possible. For that matter if we sell 2 calls at 75 and buy 1 call at 70 and 1 call at 80, the pay off from short and long calls will exactly cancel each other and the pay out at initiation and expiry will be zero. Hence the options seems to be fairly priced with no arbitrage possible

8. a. At index level of 1250, investing 250000 is like buying 200 units of the index. Since the S&P mini pays $50 for each point move in the index, to simulate 250000 investment or 200 units of S&P, we should go long in 4 contracts of S&P mini futures. Each contract will pay equaivalent of $50 per unit move in S&P and 4 contracts will give $200 per unit move in S&P which is same as buying 200 units of S&P in cash .

b. To fully hedge a portfolio with beta of 1.2, we should get futures which will mimick the move of the 1.2 times of the portfolio move i.e is (550000 * 1.2) = $ 660,000. At 660,000 investment in S&P index is like (660000/1250) = 528 units of the index. 528 units of index will correspond to (528/50) = 10.56 S&P Emini contracts. Hence to fully hedge, we should short 10.56 S&P E mini contracts

c. We saw above that $550000 portfolio with beta 1.2 is like $ 660000 S&P index portolio which has beta 1. Now if we want to hedge such that the resultant portfolio has beta 0.5, then we should short half of $660000 since the beta is 1 for the market index i.e short index equivalent to $ 330000 which means 5.28 S&P E mini contracts should be shorted . (this level is half of level calculated in part b above)

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