Investments, tenth edition

  

  In General  

  Our Numbers  

  Stock 

value 

  S  

 T 

   1   D  

  S  

 T 

   1  5 

    2   Payback of loan  

    2  (  X     1     D  )  

        2  75  

        TOTAL  

  S  

 T 

   2   X  

  S  

 T 

   2  70 

where  S  

 T 

  denotes the stock price at the option expiration date. Notice that the payoff to the 

stock is the ex-dividend stock value plus dividends received. Whether the total payoff to 

the stock-plus-borrowing position is positive or negative depends on whether  S  

 T 

  exceeds  X.  

The net cash outlay required to establish this leveraged equity position is  S  

0

   2  $71.51,  or, 

more generally,  S  

0

   2  ( X   1   D )/(1  1   r  

 f 

 ) 

 T 

 , that is, the current price of the stock,  S  

0

 , less the 

initial cash inflow from the borrowing position. 

 The payoff to the call option will be  S  

 T 

   2   X  if the option expires in the money and zero 

otherwise. Thus the option payoff is equal to the leveraged equity payoff when that payoff 

is positive and is greater when the leveraged equity position has a negative payoff. Because 

the option payoff is always greater than or equal to that of the leveraged equity position, 

the option price must exceed the cost of establishing that position. 

 Therefore, the value of the call must be greater than  S  

0

   2  ( X   1   D )/(1  1   r  

 f 

 ) 

 T 

 , or, more 

generally,   

C $ S

0

2 PV(X) 2 PV(D) 

where PV( X ) denotes the present value of the exercise price and PV( D ) is the present value 

of the dividends the stock will pay at the option’s expiration. More generally, we can inter-

pret PV( D ) as the present value of any and all dividends to be paid prior to the option expi-

ration date. Because we know already that the value of a call option must be nonnegative, 

we may conclude that  C  is greater than the  maximum  of either 0 or  S  

0

   2  PV( X )  2  PV( D ). 

 We also can place an upper bound on the possible value of the call; this bound is simply 

the stock price. No one would pay more than  S  

0

  dollars for the right to purchase a stock 

currently worth  S  

0

  dollars. Thus  C  #  S  

0

 . 

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Final PDF to printer

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  C H A P T E R  

2 1

 Option 

Valuation

727

  Figure  21.2   demonstrates  graphically 

the range of prices that is ruled out by these 

upper and lower bounds for the value of a call 

option. Any option value outside the shaded 

area is not possible according to the restric-

tions we have derived. Before expiration, the 

call option value normally will be  within   the 

allowable range, touching neither the upper 

nor lower bound, as in  Figure 21.3 .