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benefiting the call option holder and increasing the option value. As a corollary to this issue,
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

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