Investments, tenth edition

A: American Put Value

B: European Put Value

X

X

S*

X

X

PV(X)

S

0

S

0 

PV(X)

Put Value

Time

Value

 Figure 21.4 

Put option values as a function of the current stock price  

 In light of this discussion, explain why the put-call parity relationship is valid only for European options on 

non-dividend-paying stocks. If the stock pays no dividends, what  inequality  for American options would 

correspond to the parity theorem? 

 CONCEPT CHECK 

21.2 

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  C H A P T E R  

2 1

 Option 

Valuation 

729

    21.3 

Binomial Option Pricing 

   Two-State Option Pricing 

 A complete understanding of commonly used option-valuation formulas is difficult with-

out a substantial mathematics background. Nevertheless, we can develop valuable insight 

into option valuation by considering a simple special case. Assume that a stock price can 

take only two possible values at option expiration: The stock will either increase to a given 

higher price or decrease to a given lower price. Although this may seem an extreme simpli-

fication, it allows us to come closer to understanding more complicated and realistic mod-

els. Moreover, we can extend this approach to describe far more reasonable specifications 

of stock price behavior. In fact, several major financial firms employ variants of this simple 

model to value options and securities with optionlike features. 

 Suppose the stock now sells at  S  

0

   5  $100, and the price will either increase by a factor 

of  u   5  1.20 to $120 ( u  stands for “up”) or fall by a factor of  d   5  .9 to $90 ( d  stands for 

“down”) by year-end. A call option on the stock might specify an exercise price of $110 

and a time to expiration of 1 year. The interest rate is 10%. At year-end, the payoff to the 

holder of the call option will be either zero, if the stock falls, or $10, if the stock price goes 

to $120. 

 These possibilities are illustrated by the following value “trees”:     

 120 

10

 100 

C

 

  90 

  0

 

Stock price 

Call option value

 Compare the payoff of the call to that of a portfolio consisting of one share of the 

stock and borrowing of $81.82 at the interest rate of 10%. The payoff of this portfolio also 

depends on the stock price at year-end:

   Value of stock at year-end 

 $90  

$120 

    2   Repayment of loan with interest  

      2  90  

      2  90  

      TOTAL  

 $   0 

 $   30 

We know the cash outlay to establish the portfolio is $18.18: $100 for the stock, less the 

$81.82 proceeds from borrowing. Therefore the portfolio’s value tree is     

 30

 18.18

  

 

0

 The payoff of this portfolio is exactly three times that of the call option for either value 

of the stock price. In other words, three call options will exactly replicate the payoff to the 

portfolio; it follows that three call options should have the same price as the cost of estab-

lishing the portfolio. Hence the three calls should sell for the same price as this  replicating 

portfolio.   Therefore, 

  3C 5 $18.18

 or each call should sell at  C   5  $6.06. Thus, given the stock price, exercise price, interest 

rate, and volatility of the stock price (as represented by the spread between the up or down 

movements), we can derive the fair value for the call option. 

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730 

P A R T   V I

  Options, Futures, and Other Derivatives

 This valuation approach relies heavily on the notion of replication. With only two pos-

sible end-of-year values of the stock, the payoffs to the levered stock portfolio replicate the 

payoffs to three call options and, therefore, command the same market price. Replication is 

behind most option-pricing formulas. For more complex price distributions for stocks, the 

replication technique is correspondingly more complex, but the principles remain the same. 

 One way to view the role of replication is to note that, using the numbers assumed for 

this example, a portfolio made up of one share of stock and three call options written is 

perfectly hedged. Its year-end value is independent of the ultimate stock price:

  Stock 

value 

 

$90  

$120 

    2   Obligations from 3 calls written  

    2  0  

    2  30  

   Net 

payoff 

 $90 

 $  90 

The investor has formed a riskless portfolio, with a payout of $90. Its value must be the 

present value of $90, or $90/1.10  5  $81.82. The value of the portfolio, which equals $100 

from the stock held long, minus 3 C  from the three calls written, should equal $81.82. 

Hence $100  2  3 C   5  $81.82, or  C   5  $6.06. 

 The ability to create a perfect hedge is the key to this argument. The hedge locks in the 

end-of-year payout, which therefore can be discounted using the  risk-free  interest rate. To 

find the value of the option in terms of the value of the stock, we do not need to know either 

the option’s or the stock’s beta or expected rate of return. The perfect hedging, or replica-

tion, approach enables us to express the value of the option in terms of the current value 

of the stock without this information. With a hedged position, the final stock price does 

not affect the investor’s payoff, so the stock’s risk and return parameters have no bearing. 

 The hedge ratio of this example is one share of stock to three calls, or one-third. This 

ratio has an easy interpretation in this context: It is the ratio of the range of the values of the 

option to those of the stock across the two possible outcomes. The stock, which originally 

sells for  S  

0

   5  100, will be worth either  d   3  $100  5  $90 or  u   3  $100  5  $120, for a range of 

$30. If the stock price increases, the call will be worth  C  

 u 

   5  $10, whereas if the stock price 

decreases, the call will be worth  C  

 d 

   5  0, for a range of $10. The ratio of ranges, 10/30, is 

one-third, which is the hedge ratio we have established. 

 The hedge ratio equals the ratio of ranges because the option and stock are perfectly 

correlated in this two-state example. Because they are perfectly correlated, a perfect hedge 

requires that they be held in a fraction determined only by relative volatility. 

 We can generalize the hedge ratio for other two-state option problems as 

  H 5

C

u

2 C

d

uS

0

2 dS

0

 

where  C  

 u 

  or  C  

 d 

  refers to the call option’s value when the stock goes up or down, respec-

tively, and  uS  

0

   and   dS  

0

  are the stock prices in the two states. The hedge ratio,  H,   is  the 

ratio of the swings in the possible end-of-period values of the option and the stock. If the 

investor writes one option and holds  H  shares of stock, the value of the portfolio will be 

unaffected by the stock price. In this case, option pricing is easy: Simply set the value of 

the hedged portfolio equal to the present value of the known payoff. 

 Using our example, the option-pricing technique would proceed as follows:

    1.  Given the possible end-of-year stock prices,  uS  

0

   5  120 and  dS  

0

   5  90, and the 

exercise price of 110, calculate that  C  

 u 

   5  10 and  C  

 d 

   5  0. The stock price range 

is 30, while the option price range is 10.  

   2.  Find that the hedge ratio of 10/30  5  

1

⁄

3

.  

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  C H A P T E R  

2 1

 Option 

Valuation 

731

   3.  Find that a portfolio made up of 

1

⁄

3

 share with one written option would have an 

end-of-year value of $30 with certainty.  

   4.  Show that the present value of $30 with a 1-year interest rate of 10% is $27.27.  

   5.  Set the value of the hedged position to the present value of the certain payoff: 

    

1

@

3

 

S

0

2 C

0

5 $27.27

 $33.33 2 C

0

5 $27.27     

 6.  Solve for the call’s value,  C  

0

   5  $6.06.    

 What if the option is overpriced, perhaps selling for $6.50? Then you can make arbi-

trage profits. Here is how: