Investments, tenth edition

21.3 

  Generalizing the Two-State Approach 

 Although the two-state stock price model seems simplistic, we can generalize it to incor-

porate more realistic assumptions. To start, suppose we were to break up the year into 

two 6-month segments, and then assert that over each half-year segment the stock price 

could take on two values. In this example, we will say it can increase 10% (i.e.,  u   5  1.10) 

or decrease 5% (i.e.,  d   5  .95). A stock initially selling at 100 could follow these possible 

paths over the course of the year:     

 

 121

 110 

 100 

104.50

  

 

95

  

 

90.25

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732 

P A R T   V I

  Options, Futures, and Other Derivatives

The midrange value of 104.50 can be attained by two paths: an increase of 10% followed 

by a decrease of 5%, or a decrease of 5% followed by a 10% increase. 

 There are now three possible end-of-year values for the stock and three for the option:     

 

C

uu

 

C

u

 C 

C

ud

 5 C

du

 

C

d

 

C

dd

 Using methods similar to those we followed above, we could value  C  

 u 

  from knowl-

edge of  C  

 uu 

  and  C  

 ud 

 , then value  C  

 d 

  from knowledge of  C  

 du 

  and  C  

 dd 

 , and finally value  C  

from knowledge of  C  

 u 

  and  C  

 d 

 . And there is no reason to stop at 6-month intervals. We 

could next break the year into four 3-month units, or twelve 1-month units, or 365 1-day 

units, each of which would be posited to have a two-state process. Although the calcula-

tions become quite numerous and correspondingly tedious, they are easy to program into 

a computer, and such computer programs are used widely by participants in the options 

market.  

 

 

 Suppose that the risk-free interest rate is 5% per 6-month period and we wish to 

value a call option with exercise price $110 on the stock described in the two-period 

price tree just above. We start by finding the value of  C  

 u 

 . From this point, the call can 

rise to an expiration-date value of  C  

 uu 

   5  $11 (because at this point the stock price is 

 u   3   u   3   S  

0

   5  $121) or fall to a final value of  C  

 ud 

   5  0 (because at this point, the stock 

price is  u   3   d   3   S  

0

   5  $104.50, which is less than the $110 exercise price). Therefore the 

hedge ratio at this point is

   H 5

C

uu

2

C

ud

uuS

0

2

udS

0

5

$11 2 0

$121 2 104.50

5

2

3

 

Thus, the following portfolio will be worth $209 at option expiration regardless of the 

ultimate stock price: