Investments, tenth edition

Cash Flow in 1 Year for 

Each Possible Stock Price

Initial Cash Flow

S 5 90

S 5 120

Buy 3 options

2

16.50

0

30

Short-sell 1 share; repay in 1 year

100

2

90

2

120

Lend $83.50 at 10% interest rate

2

83.50

91.85

91.85

 TOTAL

0

1.85

1.85

    

The riskless cash flow in 1 year per option is $1.85/3   

5 

  $.6167, and the present value is 

$.6167/1.10  5  $.56, precisely the amount by which the option is underpriced.  

   4.      a.    C  

 u 

   2   C  

 d 

   5  $6.984  2  0  

   b.    uS  

0

   2   dS  

0

   5  $110  2  $95  5  $15  

   c.  6.984/15  

5  .4656  

   d.   

  

The portfolio must have a market value equal to the present value of $44.232.    

 e.  $44.232/1.05  

5  $42.126  

   f.  .4656  

3  $100  2   C  

0

   5  $42.126  

 

   C  

0

   5  $46.56  2  $42.126  5  $4.434     

   5.  When  Δ t  shrinks, there should be lower possible dispersion in the stock price by the end of 

the subperiod because each shorter subperiod offers less time in which new information can 

move stock prices. However, as the time interval shrinks, there will be a correspondingly greater 

number of these subperiods until option expiration. Thus,  total  volatility over the remaining life 

of the option will be unaffected. In fact, take another look at  Figure 21.2 . There, despite the fact 

that  u   and   d  each get closer to 1 as the number of subintervals increases and the length of each 

subinterval falls, the total volatility of the stock return until option expiration is unaffected.  

   6.  Because   s   5  .6,  s  

2

   5  .36. 

   d

1

5

ln(100/95) 1 (.10 1 .36/2).25

.6

".25

5 .4043

 d

2

5 d

1

2 .6

".25 5 .1043

  

     Using  Table 21.2  and interpolation, or from a spreadsheet function: 

   N(d

1

) 5 .6570

 N(d

2

) 5 .5415

 C 5 100 3 .6570 2 95e

2.103.25

3 .5415 5 15.53     

 7.  Implied volatility exceeds .2783. Given a standard deviation of .2783, the option value is $7. A 

higher volatility is needed to justify an $8 price. Using Spreadsheet 21.1 and Goal Seek, you can 

confirm that implied volatility at an option price of $8 is .3138.  

   8.  A $1 increase in stock price is a percentage increase of 1/122  5  .82%. The put option will fall by 

(.4  3  $1)  5  $.40, a percentage decrease of $.40/$4  5  10%. Elasticity is  2 10/.82  5   2 12.2.  

   9.  The delta for a call option is  N ( d  

1

 ), which is positive, and in this case is .547. Therefore, for every 

10 option contracts purchased, you would need to  short   547  shares  of  stock.