Investments, tenth edition

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766 

P A R T   V I

  Options, Futures, and Other Derivatives

   50.  You build a binomial model with one period and assert that over the course of a year, the stock 

price will either rise by a factor of 1.5 or fall by a factor of 2/3. What is your implicit assumption 

about the volatility of the stock’s rate of return over the next year?  

 

 

 

51.  Use the put-call parity relationship to demonstrate that an at-the-money call option on a 

nondividend-paying stock must cost more than an at-the-money put option. Show that the prices 

of the put and call will be equal if  S   5  (1  1   r ) 

 T 

 .  

   52.  Return to Problem 36. Value the call option using the risk-neutral shortcut described in the 

box on page 736. Confirm that your answer matches the value you get using the two-state 

approach.  

   53.  Return to Problem 38. What will be the payoff to the put,  P  

 u 

 , if the stock goes up? What will 

be the payoff,  P  

 d 

 , if the stock price falls? Value the put option using the risk-neutral shortcut 

described in the box on page 736. Confirm that your answer matches the value you get using the 

two-state  approach.    

            1.  T h e  board  of  directors  of  Abco  Company  is  concerned  about  the  downside  risk  of  a  $100 

million equity portfolio in its pension plan. The board’s consultant has proposed temporarily (for 

1 month) hedging the portfolio with either futures or options. Referring to the following table, the 

consultant states:

     a. 

 “The $100 million equity portfolio can be fully protected on the downside by selling 

(shorting) 4,000 futures contracts.”  

    b.   “The cost of this protection is that the portfolio’s expected rate of return will be zero percent.” 

Market, Portfolio, and Contract Data

Equity index level

99.00

Equity futures price

100.00

Futures contract multiplier

$250

Portfolio beta

1.20

Contract expiration (months)

3

     Critique the accuracy of each of the consultant’s two statements.     

   2.  Michael Weber, CFA, is analyzing several aspects of option valuation, including the determinants 

of the value of an option, the characteristics of various models used to value options, and the 

potential for divergence of calculated option values from observed market prices.

     a.   What is the expected effect on the value of a call option on common stock if the volatility of 

the underlying stock price decreases? If the time to expiration of the option increases?  

    b.   Using the Black-Scholes option-pricing model, Weber calculates the price of a 3-month call 

option and notices the option’s calculated value is different from its market price. With respect 

to Weber’s use of the Black-Scholes option-pricing model,

     i.  Discuss why the calculated value of an out-of-the-money European option may differ 

from its market price.  

   ii.  Discuss why the calculated value of an American option may differ from its market 

price.        

   3.  Joel Franklin is a portfolio manager responsible for derivatives. Franklin observes an American-

style option and a European-style option with the same strike price, expiration, and underly-

ing stock. Franklin believes that the European-style option will have a higher premium than the 

American-style option.

     a.   Critique Franklin’s belief that the European-style option will have a higher premium. Franklin 

is asked to value a 1-year European-style call option for Abaco Ltd. common stock, which last 

traded at $43.00. He has collected the information in the following table.  

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  C H A P T E R  

2 1

 Option 

Valuation 

767

 Closing stock price 

 $43.00 

 Call and put option exercise price 

 45.00 

 1-year put option price 

 4.00 

 1-year Treasury bill rate 

 5.50% 

 Time to expiration 

 One year 

    b.   Calculate, using put-call parity and the information provided in the table, the European-style 

call option value.  

    c.   State the effect, if any, of each of the following three variables on the value of a call option. 

(No calculations required.)

     i.  An increase in short-term interest rate.  

   ii.  An increase in stock price volatility.  

   iii.  A  decrease  in  time  to  option  expiration.        

   4.  A stock index is currently trading at 50. Paul Tripp, CFA, wants to value 2-year index options using 

the binomial model. The stock will either increase in value by 20% or fall in value by 20%. The annual 

risk-free interest rate is 6%. No dividends are paid on any of the underlying securities in the index.

     a.   Construct a two-period binomial tree for the value of the stock index.  

    b.   Calculate the value of a European call option on the index with an exercise price of 60.  

    c.   Calculate the value of a European put option on the index with an exercise price of 60.  

    d.   Confirm that your solutions for the values of the call and the put satisfy put-call parity.     

   5.  Ken Webster manages a $200 million equity portfolio benchmarked to the S&P 500 index. 

Webster believes the market is overvalued when measured by several traditional fundamental/

economic indicators. He is concerned about potential losses but recognizes that the S&P 500 

index could nevertheless move above its current 1136 level. 

     Webster is considering the following  option collar   strategy:

    

•

  Protection for the portfolio can be attained by purchasing an S&P 500 index put with a strike 

price of 1130 (just out of the money).  

   

•

  The put can be financed by selling two 1150 calls (farther out-of-the-money) for every put purchased.  

   

•

  Because the combined delta of the two calls (see following table) is less than 1 (that is, 

2  3  .36  5  .72), the options will not lose more than the underlying portfolio will gain if the 

market advances.     

The information in the following table describes the two options used to create the collar.

Characteristics

1150 Call

1130 Put

Option price

$8.60

$16.10

Option implied volatility

22%

24%

Option’s delta

0.36

2

0.44

Contracts needed for collar

602

301

Notes:

•

  Ignore transaction costs.

•

  S&P 500 historical 30-day volatility 5 23%.

•

  Time to option expiration 5 30 days.

     a.   Describe the potential returns of the combined portfolio (the underlying portfolio plus the 

option collar) if after 30 days the S&P 500 index has:

     i.  risen approximately 5% to 1193.  

   ii.  remained at 1136 (no change).  

   iii.  declined by approximately 5% to 1080. 

     (No  calculations  are  necessary.)       

  b.   Discuss the effect on the hedge ratio (delta) of  each  option as the S&P 500 approaches the 

level for  each  of the potential outcomes listed in part ( a ).  

    c.   Evaluate the pricing of  each  of the following in relation to the volatility data provided:

     i.  the  put  

    ii.  the  call          

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768 

P A R T   V I

  Options, Futures, and Other Derivatives

 E-INVESTMENTS EXERCISES 

 Select a stock for which options are listed on the CBOE Web site (  www.cboe.com  ). The 

price data for captions can be found on the “delayed quotes” menu option. Enter a ticker 

symbol for a stock of your choice and pull up its option price data. 

 Using daily price data from   finance.yahoo.com   calculate the annualized standard 

deviation of the daily percentage change in the stock price. Create a Black-Scholes 

option-pricing model in a spreadsheet, or use our Spreadsheet 21.1, available at   www.