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Unformatted text preview: BUS 415: Investment and Portfolio Management SPRING 2011 PROBLEM SET 7 (Team work) 1. ( 8 points ) Consider stock XYZ whose prices evolve according to a binomial process. The stock is currently trading at $50 and it is known that at the end of one year, the price will either increase by 20% (in the “up” state) or decrease by 20% (in the “down” state). Next, consider a call option on stock XYZ which expires at the end of one year and has a strike price of $45. Furthermore, assume that the riskfree interest rate is 5% per annum with continuous compounding. A. (4 points) Construct a riskless portfolio (P) using the call option and the given stock XYZ. Indicate the number of stocks and the number of call options in the riskless portfolio (Hint : a riskless portfolio should have the same payoffs) B. (2 points) What are the payoffs of portfolio P in the two states? C. (2 points) What should be the current price of portfolio P? Solution: 1. Construct a riskless portfolio (P) based on the call option and the given stock XYZ. S(up)= $60 $15 So = $50 4 3 40 60 15 ) dn ( S ) up ( S ) dn ( f ) up ( f = = = ∆ S(dn)= $40 $0 Stock Call The riskless portfolio is: Long (Stock)  4 3 = ∆ unit, Short (Call)  1 unit. 2. What are the payoffs of portfolio P in the two states? Net payoff of the portfolio in the “up” state is: 30 $ ) 15 ( x 1 ) 60 ( x 4 3 = Net payoff of the portfolio in the “down” state is: 30 $ ) ( x 1 ) 40 ( x 4 3 = 3. What should be the current price of portfolio P? The portfolio price P = 14 . 28 $ e 30 $ Xe 1 x 05 . 1 x 05 . = = , where $30 is the weighted average of portfolio payoffs. 1 2. ( 15 points ) The following three securities are trading in an economy: • Stock XYZ: its prices evolve according to a binomial process. The stock is currently trading at $50 and it is known that at the end of one year, the price will...
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 Spring '11
 john
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