1 EXAM INTRODUCTION TO FINANCIAL MATHEMATICS MTH 457-001 SPRING 2012 PROF. GABOR FRANCSICS D310 WELLS HALL, 353-7962, [e-mail] DEADLINE: 12:30PM Wednesday, February 29, 2012
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There are eight problems. These problems are similar to homework problems, problems in the textbooks, and problems solved in lecture. Write down the details (all the details)! Use Excel or Mathematica for numerical calculations, attach printout!
Problem Points 1. 2. 3. 4. 5. 6. 7. 8. Total:
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PROBLEM 1. [5 points]. What is the difference between taking a long European call position and buying the stock? What is the difference between a forward contract, future contract, and options? What is an arbitrage opportunity? Give a detailed explanation (2-3 paragraphs long).
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PROBLEM 2. [5 points]. Suppose that you write a put option contract on 100 IBM shares with strike price $110 and an expiration date in three months. The current price of IBM stock is $111. What have you committed yourself to? How much could you gain or lose?
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PROBLEM 3. [10 points] Explain a portfolio that would generate the given graph as the payoff function of the portfolio. Give a formula for the payoff function. (a)
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70
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(b)
30
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PROBLEM 4. [5 points]. Three call options on a stock have the same expiration date and strike prices of $25, $33, and $35. The prices of the options are $3, $1.2, and $1, respectively. An investor buys two 25-strike calls, sells ten 33-strike calls, and buy eight 35-strike calls. Construct the profit graph of the investor's portfolio. For what range of the stock prices would this asymmetric butterfly spread lead to a loss?
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PROBLEM 5. [5 points]. Let X be a standard normal variable. (a) Show that E(e4X ) = e8 .
(b) Calculate the probability density function of the random variable 2e4X .
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PROBLEM 6. [10 points]. The financial wizard Goldentouch Jane wants to sell you a portfolio of European options. The portfolio contains a long position in a 20-strike call, a long position in a 30-strike put, a short position in a 30-strike call, and a short position in a 20-strike put. The expiration date of the options are six months from now. The risk free interest rate is 5% per annum. What is the fair price for this portfolio? Why?
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PROBLEM 7. [5 points] The given graph is the payoff function of a portfolio consisting of European options.
E2 - E1
E1
E2
Which trading strategy among the following four has the same payoff function? (1) Buying a put option with with strike price E1 , expiration date T ; and selling a put option with strike price E2 expiration date T , E2 < E1 . (2) Buying a call option with with strike price E1 , expiration date T ; and buying a call option with with strike price E2 , expiration date T ; and selling two call options with strike price E1 +E2 expiration date T , E2 > E1 . 2 (3) Buying a call option with with strike price E1 , expiration date T and selling a call option with strike price E2 expiration date T , E2 > E1 . (4) Buying a call option with with strike price E2 , expiration date T and buying a put option with strike price E1 expiration date T , E2 > E1 . Compute the four payoff functions and their graphs.
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PROBLEM 8. [5 points]. An investor believes that there will be a big increase in a stock price. Which trading strategy among the four in Problem 7 would you recommend to the investor? Another investor believes that there will be a big change in a stock price. Which trading strategy among the four in Problem 7 would you recommend to this investor? A third investor believes that there will be no significant change in a stock price. Which trading strategy among the four in Problem 7 would you recommend to this investor? Explain your answer.
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