Problem set: 4 Course: M339D/M389D - Intro to Financial Math Page: 1 of 5 University of Texas at Austin Problem Set #4 Problem 4.1. The current price of a non-dividend-paying stock is $80 per share. You observe that the price of a three-month, at-the-money European put option on this stock equals $2.50. The continuously compounded, risk-free interest rate is 0.08. Find the premium of the European three-month, at-the-money call option on the same underlying asset. (a) About $3.08 (b) About $4.08 (c) About $4.75 (d) About $5.46 Problem 4.2. Roger owns a cow named Elsie. Her estimated worth today is $3,750. Roger enters into a forward agreement with Harry to sell him Elsie the cow in 6 months for $4,000. On the delivery date, Roger changes his mind and wants cash settlement instead. Harry agrees. They look into the Bovine Blue Book and realize that Elsie s worth on that date is $3,500. What is the cash flow that has to take place as part of the cash settlement? (a) $500 from Roger to Harry (b) $500 from Harry to Roger (c) $250 from Roger to Harry (d) $250 from Harry to Roger Problem 4.3. (5 points) A company projects to be able to a dividend of $2.00 per share at the end of the next quarter. The following dividend payments will be increasing by five cents. Its stock price today is $100.00. Assume that
Problem set: 4 Course: M339D/M389D - Intro to Financial Math Page: 2 of 5 the continuously compounded interest rate equals 4%. What is the prepaid forward price for a 6 month prepaid forward contract on the above stock with delivery immediately after the second dividend? (a) $100 (b) $95.96 (c) $96.01 (d) $93.82 Problem 4.4. A customer buys a six-month at-the-money put on an index when the market price of the index is 50. The premium for the put is 2. The continuously compounded, risk-free interest rate equals 0.06. The price of the index at expiration is modeled as follows 45, with probability 0.6, 50, with probability 0.3, 55, with probability 0.1. What is the expected value of the profit of the long put? (a) $0.53 (b) $0.97 (c) $1.03 (d) $1.12 Problem 4.5. We are given the following European-call prices for options on the same underlying asset: $50-strike $11 $55-strike $6 $60-strike $4
Problem set: 4 Course: M339D/M389D - Intro to Financial Math Page: 3 of 5 Assume that the continuously compounded, risk-free interest rate is strictly positive. Which of the following portfolios would exploit an arbitrage opportunity stemming from the above stock prices? (a) The call bear spread only. (b) The call bull spread only. (c) Both the call bull and the call bear spread. (d) Neither the call bull or call bear spread, but there is an arbitrage opportunity. (e) There is no apparent arbitrage opportunity. Problem 4.6. (5 points) The future value in one year of the total aggregate costs of manufacturing a widget is $100. You will sell a widget in one year at its market price of S(1). Assume that the continuously compounded, risk-free interest rate equals 5%. You purchase a one-year, $120-strike put on one widget for a premium of $7. You sell some of the potential gain by writing a one-year, $150-strike call on one widget for a $3 premium. What is the range of the profit of your total hedged porfolio? (a) [14.20, 44.20] (b) [14.75, 44.75] (c) [15.79, 45.79] (d) [120, 150] Problem 4.7. (5 points) The current futures price for delivery in three years equals $100. You use a twoperiod binomial tree to model the evolution of the futures price over the following year for the purposes of pricing a one-year, $105-strike European call option on the futures contract. To be able to construct the tree, you are given the following information: u F /d F = 4/3 where u F denotes the up factor in the futures-price tree and d F denotes the down factor in the futures-price tree. The risk-neutral probability of the futures price going up in a single step is 1/2.
Problem set: 4 Course: M339D/M389D - Intro to Financial Math Page: 4 of 5 The continuously-compounded, risk-free interest rate equals 0.04. What is the price of the above European call option? (a) About $5.76 (b) About $6.15 (c) About $7.15 (d) About $8.17 Problem 4.8. HAW, Inc. plans to pay a $1.10 dividend per share in 3 months and a $1.15 dividend in 6 months. HAWs share price today is $45.60 and the continuously compounded interest rate is 8.4%. What is the price of a forward contract with delivery immediately after the second dividend? (a) $45.28 (b) $45.96 (c) $45.60 (d) $46.24 (e) None of the above Problem 4.9. (5 points) Let the current price of a non-dividend-paying stock be $95 per share. The price of this stock in one year is modeled by a one-period binomial model. The two possible prices that the stock can attain in this model are $120 and $80. Assume that the continuously compounded risk-free interest rate equals 0.05. An investor purchases a $100-strike straddle on the above stock with the exercise date in one year. What is the initial cost of this position? (a) About 16.26 (b) About 17.28 (c) About 18.30 (d) About 19.02
Problem set: 4 Course: M339D/M389D - Intro to Financial Math Page: 5 of 5 Problem 4.10. Source: Sample FM(DM) Problem #5. A market index has the following characteristics: One share of the PS index currently sells for 1,000. The PS index does not pay dividends. Sam wants to lock in the ability to buy this index in one year for a price of $1,025. He can do this by buying or selling European put and call options with a strike price of $1,025. The annual effective risk-free interest rate is 5%. Determine which of the following gives the hedging strategy that will achieve Sam s objective and also gives the cost today of establishing this position. (a) Buy the call and sell the put, spend 23.81. (b) Buy the call and sell the put, receive 23.81. (c) Buy the put and sell the call, no cost. (d) Buy the put and sell the call, spend 23.81. (e) Buy the put and sell the call, receive 23.81.