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Sample Midterm Exam #3--Answers Fi 4000—Fall 2008 Problem 1 (25 points) USB stock currently trades for \$80 per share. Each year, there are two possible outcomes. The stock price will either increase by 20% or decrease by 10%. The risk-free interest rate is 5% per year. You would like to find the fair value of a European put option on USB stock which expires in one year and has an exercise (strike) price of \$85. a. Draw binomial trees to illustrate the price of USB stock and the value of \$1 invested at the riskless rate of return. [5 points] b. Calculate the no-arbitrage price of the European put option. [8 points] First, from the stock prices we can write down the payoffs to the option at maturity: To find P 0 , the current price of the put option, we first solve for “state prices” q u and q d , where q u is the price today of a security that pays off \$1 if the stock price goes up (and pays \$0 if the stock price goes down), and q d is the price today of a security that pays off \$1 if the stock price goes down (and pays \$0 if the stock price goes up). Now, these state prices must be consistent with both the current stock and the bond prices (see the trees in part a.), so we have d d u q q q 05 . 1 1.05q 1 72 96 80 u + = + = 1 S d = 72 S u = 96 S 0 = 80 B d = 1.05 B u = 1.05 B 0 = 1 P d = max[0,85-72]=13 P 0 = ? P u = max[0,85-96] = 0

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Solving these two equations we get 47625 . 0 47615 . 0 = = d u q q Then we can calculate the European put value using the state prices and the payoffs at maturity: 19 . 6 \$ 47625 . 0 * 13 * 13 * 0 = = + = d u u q q P c. What would be the price today of an American put option with strike price 85 and 12 months to maturity? [6 points] Note that the most we can get from exercising an American call today is \$85 – 80 = 5. This is less than what we could get from selling a European call! (part b. above). Therefore, we will not exercise the call today, and hence it is valued as an European call option with value we found above, \$6.19. d. What would be the fair market price today of a European call option with strike price 80 and 12 months till maturity? [6 points] We can use the state prices from above to price the call option: 62 . 7 \$ 47615 . 0 * 16 * 0 * 16 = = + = d u u q q C 2 C d = max[0,80-72]=0 C 0 = ? C u = max[0,96-80] = 16
Problem 2 (20 points) 1. Suppose that stock XYZ is currently trading at \$75 per share, and a European call option with strike price \$70 and a 12 month maturity is valued at \$11. The stock is not expected to pay dividends in the near future. Furthermore, suppose that the annual risk-free rate is 6%.

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