1.A box spread is a combination of a bull spread composed of two call options with strike prices 1Xand 2Xand a bear spread composed of two put options with the same two strike prices. a)Describe the payoff from a box spread on the expiration date of the options. b)What would be a fair price for the box spread today? Define variables as necessary.c)Under what circumstances might an investor choose to construct a box spread? d)What sort of investor do you think is most likely to invest in such an option combination, i.e. a hedger, speculator or arbitrageur? Explain your answer.2. Form a long butterfly spread using the three call options in the table below.C1X = $90T = 180 daysC2X = $100T = 180 daysC3X = $110T = 180 daysPrice16.330010.30006.0600DELTA0.78600.61510.4365GAMMA0.01380.01810.0187THETA-11.2054-12.2607-11.4208VEGA20.461926.841627.6602RHO30.708525.251518.5394a)What does it cost to establish the butterfly spread? b)Calculate each of the Greek measures for this butterfly spread position and explain how each can be interpreted. c)How would you make this option portfolio delta neutral? What would be achieved by doing so? d)Suppose that tomorrow the price of C1 falls to $12.18 while the prices of C2 and C3 remain the same. Does this create an arbitrage opportunity? Explain.3.Consider a six month American put option on index futures where the current futures price is 450, the exercise price is 450, the risk-free rate of interest is 7 percent per annum, the continuous dividend yield of the index is 3 percent, and the volatility of the index is 30 percent per annum. The futures contract underlying the option matures in seven months. Using a three-step binomial tree, calculatea)the price of the American put option now, b)the delta of the option with respect to the futures price, c)the delta of the option with respect to the index level, and d)the price of the corresponding European put option on index futures. e)Apply the control variate technique to improve your estimate of the American option price andof the delta of the option with respect to the futures price. Note that the Black-Scholes price of the European put option is $36.704 and the delta with respect to the futures price given by Black-Scholes is –0.442.106

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