call option values are higher when interest rates rise (holding the stock price constant), be-

cause higher interest rates also reduce the present value of the exercise price.

Finally, the dividend payout policy of the firm affects option values. A high dividend pay-

out policy puts a drag on the rate of growth of the stock price. For any expected total rate of

return on the stock, a higher dividend yield must imply a lower expected rate of capital gain.

This drag on stock appreciation decreases the potential payoff from the call option, thereby

lowering the call value. Table 15.1 summarizes these relationships.

>

2. Prepare a table like Table 15.1 for the determinants of put option values. How

Concept

should put values respond to increases in S, X, T, , rf , and dividend payout?

CHECK

15.2 BINOMIAL OPTION PRICING

Two-State Option Pricing

A complete understanding of commonly used option valuation formulas is difficult without

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

option valuation by considering a simple special case. Assume 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 simplification, it allows

us to come closer to understanding more complicated and realistic models. 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 $100, and the price will either double to $200 or fall in half

to $50 by year-end. A call option on the stock might specify an exercise price of $125 and a

If This Variable Increases The Value of a Call Option

TA B L E 15.1

Stock price, S Increases

Determinants of

Exercise price, X Decreases

call option values

Volatility, Increases

Time to expiration, T Increases

Interest rate, rf Increases

Dividend payouts Decreases

Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

535

15 Option Valuation

time to expiration of one year. The interest rate is 8%. At year-end, the payoff to the holder of

the call option will be either zero, if the stock falls, or $75, if the stock price goes to $200.

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

$200 $75

$100 C

$50 $0

Stock price Call option value

Compare this payoff to that of a portfolio consisting of one share of the stock and borrow-

ing of $46.30 at the interest rate of 8%. The payoff of this portfolio also depends on the stock

price at year-end.

Value of stock at year-end $50 $200

Repayment of loan with interest 50 50

Total $0 $150

We know the cash outlay to establish the portfolio is $53.70: $100 for the stock, less the

$46.30 proceeds from borrowing. Therefore, the portfolio™s value tree is

$150

$53.70

$0

The payoff of this portfolio is exactly twice that of the call option for either value of the stock

price. In other words, two call options will exactly replicate the payoff to the portfolio; it fol-

lows that two call options should have the same price as the cost of establishing the portfolio.

Hence, the two calls should sell for the same price as the “replicating portfolio.” Therefore

2C $53.70

or each call should sell at C $26.85. Thus, given the stock price, exercise price, interest rate,

and volatility of the stock price (as represented by the magnitude of the up or down move-

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

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

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

to two call options and so need to command the same market price. This notion of 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 two call options written is perfectly

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

Stock value $50 $200

Obligations from 2 calls written 0 150

Net payoff $50 $ 50

Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

536 Part FIVE Derivative Markets

The investor has formed a riskless portfolio with a payout of $50. Its value must be the

present value of $50, or $50/1.08 $46.30. The value of the portfolio, which equals $100

from the stock held long, minus 2C from the two calls written, should equal $46.30. Hence,

$100 2C $46.30, or C $26.85.

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

of-year payout, which 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 replication, approach enables

us to express the value of the option in terms of the current value of the stock without this in-

formation. 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 two calls, or one-half. For every

call option written, one-half share of stock must be held in the portfolio to hedge away risk.

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 option is worth either

zero or $75, for a range of $75. The stock is worth either $50 or $200, for a range of $150. The

ratio of ranges, $75/$150, is one-half, which is the hedge ratio we have established.

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

lated in this two-state example. When the returns of the option and stock are perfectly corre-

lated, a perfect hedge requires that the option and stock be held in a fraction determined only

by relative volatility.

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

C C

H

S S

where C or C refers to the call option™s value when the stock goes up or down, respectively,

and S and S 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, S $200 and S $50, and the exercise

price of $125, calculate that C $75 and C $0. The stock price range is $150,

while the option price range is $75.

2. Find that the hedge ratio is $75/$150 0.5.

3. Find that a portfolio made up of 0.5 shares with one written option would have an end-of-

year value of $25 with certainty.

4. Show that the present value of $25 with a one-year interest rate of 8% is $23.15.

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

0.5S0 C0 $23.15

$50 C0 $23.15

6. Solve for the call™s value, C0 $26.85.

What if the option were overpriced, perhaps selling for $30? Then you can make arbitrage

profits. Here is how.

Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

537

15 Option Valuation

Cash Flow in 1 Year

for Each Possible

Stock Price

Initial

Cash Flow S $50 S $200

1. Write 2 options. $ 60 $ 0 $ 150

2. Purchase 1 share. 100 50 200

3. Borrow $40 at 8% interest and repay in 1 year. 40 43.20 43.20

Total $ 0 $ 6.80 $ 6.80

Although the net initial investment is zero, the payoff in one year is positive and riskless.

If the option were underpriced, one would simply reverse this arbitrage strategy: Buy the op-

tion, and sell the stock short to eliminate price risk. Note, by the way, that the present value of

the profit to the above arbitrage strategy equals twice the amount by which the option is over-

priced. The present value of the risk-free profit of $6.80 at an 8% interest rate is $6.30. With

two options written in the strategy above, this translates to a profit of $3.15 per option, exactly

the amount by which the option was overpriced: $30 versus the “fair value” of $26.85.

<

3. Suppose the call option had been underpriced, selling at $24. Formulate the arbi- Concept

trage strategy to exploit the mispricing, and show that it provides a riskless cash

CHECK

flow in one year of $3.08 per option purchased.

Generalizing the Two-State Approach

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

more realistic assumptions. To start, suppose we were to break up the year into two six-month

segments and then assert that over each half-year segment the stock price could take on two

values. Here we will say it can increase 10% or decrease 5%. A stock initially selling at $100

could follow the following possible paths over the course of the year:

$121

$110

$100 $104.50

$95

$90.25

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 an increase of 10%.

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

C

C

C C

C

C

Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

538 Part FIVE Derivative Markets

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

C and C , then value C from knowledge of C and C , and finally value C from

knowledge of C and C . And there is no reason to stop at six-month intervals. We could next

break the year into 4 three-month units, or 12 one-month units, or 365 one-day units, each of

which would be posited to have a two-state process. Although the calculations 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 six-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

15.1 EXAMPLE above. We start by finding the value of C . From this point, the call can rise to an expiration-

date value of C $11 (since at this point the stock price is S $121) or fall to a final

Binomial

value of C 0 (since at this point the stock price is S $104.50, which is less than the

Option

$110 exercise price). Therefore, the hedge ratio at this point is

Pricing

C C 2

$11 0

H

S S 3

$121 $104.50

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

mate stock price:

S $104.50 S $121

Buy 2 shares at price S $110 $209 $242

Write 3 calls at price C 0 33

Total $209 $209

The portfolio must have a current market value equal to the present value of $209:

2 $110 3C $209/1.05 $199.047

Solve to find that C $6.984.

Next we find the value of C . It is easy to see that this value must be zero. If we reach this