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Chapter 22: Problems 3(a-d), 5(a-d), 7(a-c), 10(a-b), and 12

Chapter 22: Problem 3(a-d)

3. Assuming that a one-year call option with an exercise price of $38 is available for the stock of the DEW Corp., consider the following price tree for DEW stock over the next year:

Now

S1

S2

One Year

46.31

44.10

42

42.34

40

40.32

38.40

38.71

36.86

35.39

a. If the sequence of stock prices that DEW stock follows over the year is $40.00, $42.00, $40.32, and $38.71, describe the composition of the initial riskless portfolio of stock and options you would form and all the subsequent adjustments you would have to make to keep this portfolio riskless. Assume the one-year risk-free rate is 6 percent.

b. Given the initial DEW price of $40, what are the probabilities of observing each of the four terminal stock prices in one year? (Hint: In arriving at your answer, it will be useful to consider (1) the number of different ways that a particular terminal price could be achieved and (2) the probability of an up or down movement.)

c. Use the binomial option model to calculate the present value of this call option. d. Calculate the value of a one-year put option on DEW stock having an exercise price of$38; be sure your answer is consistent with the correct response to Part c.

Chapter 22: Problem 5(a-d)

5. Consider the following questions on the pricing of options on the stock of ARB Inc.:

a. A share of ARB stock sells for $75 and has a standard deviation of returns equal to 20 percent per year. The current risk-free rate is 9 percent and the stock pays two dividends: (1) a $2 dividend just prior to the option’s expiration day, which is 91 days from now (i.e., exactly one-quarter of a year), and (2) a $2 dividend 182 days from now (i.e., exactly one-half year). Calculate the Black-Scholes value for a European-style call option with an exercise price of $70. b. What would be the price of a 91-day European-style put option on ARB stock having the same exercise price?

c. Calculate the change in the call option’s value that would occur if ARB’s management suddenly decided to suspend dividend payments and this action had no effect on the price of the company’s stock.

d. Briefly describe (without calculations) how your answer in Part a would differ under the following separate circumstances: (1) the volatility of ARB stock increases to 30 percent, and (2) the risk-free rate decreases to 8 percent.

Chapter 22: Problem 7(a-c)

7. Suppose the current value of a popular stock index is 653.50 and the dividend yield on the index is 2.8 percent. Also, the yield curve is flat at a continuously compounded rate of 5.5 percent.

a. If you estimate the volatility factor for the index to be 16 percent, calculate the value of an index call option with an exercise price of 670 and an expiration date in exactly three months.

b. If the actual market price of this option is $17.40, calculate its implied volatility coefficient.

c. Besides volatility estimation error, explain why your valuation and the option’s traded price might differ from one another.

Chapter 22: Problem 10(a-b)

10. Melissa Simmons is the chief investment officer of a hedge fund specializing in options trading. She is currently back-testing various option trading strategies that will allow her to profit from large fluctuations—either up or down—in a stock’s price. An example of such typical trading strategy is straddle strategy that involves the combination of a long call and a long put with an identical strike price and time to maturity. She is considering the following pricing information on securities associated with Friendwork, a new Internet start-up hosting a leading online social network:

Friendwork stock: $100

Call option with an exercise price of $100 expiring in one year: $9

Put option with an exercise price of $100 expiring in one year: $8

a. Use the above information on Friendwork and draw a diagram showing the net profit/ loss position at maturity for the straddle strategy. Clearly label on the graph the break- even points of the position.

b. Melissa’s colleague proposes another lower-cost option strategy that would profit from a large fluctuation in Friendwork’s stock price:

Long call option with an exercise price of $110 expiring in one year: $6

Long put option with an exercise price of $90 expiring in one year: $5

Similar to Part a, draw a diagram showing the net profit/loss position for the above alternative option strategy. Clearly label on the graph the breakeven points of the position.

Chapter 22: Problem and 12

12. In developing the butterfly spread position, we showed that it could be broken down into two call option money spreads. Using the price data for SAS stock options from Exhibit 22.17, demonstrate how a butterfly profit structure similar to that shown in Exhibit 22.30 could be created using put options. Be specific as to the contract positions involved in the trade and show the expiration date net payoffs for the combined transaction.

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