$104.50 or $90.25; in both cases, the option will expire out of the money. (More formally,

we could note that with C C 0, the hedge ratio is zero, and a portfolio of zero

shares will replicate the payoff of the call!)

Finally, we solve for C by using the values of C and C . Concept Check 4 leads you

through the calculations that show the option value to be $4.434.

>

4. Show that the initial value of the call option in Example 15.1 is $4.434.

Concept

a. Confirm that the spread in option values is C C $6.984.

CHECK b. Confirm that the spread in stock values is S S $15.

c. Confirm that the hedge ratio is .4656 shares purchased for each call written.

d. Demonstrate that the value in one period of a portfolio comprising .4656

shares and one call written is riskless.

e. Calculate the present value of this payoff.

f. Solve for the option value.

As we break the year into progressively finer subintervals, the range of possible year-end

stock prices expands and, in fact, will ultimately take on a familiar bell-shaped distribu-

tion. This can be seen from an analysis of the event tree for the stock for a period with three

subintervals.

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Essentials of Investments, Companies, 2003

Fifth Edition

539

15 Option Valuation

S

S

S S

S S

S S

S

S

First, notice that as the number of subintervals increases, the number of possible stock

prices also increases. Second, notice that extreme events such as S or S are relatively

rare, as they require either three consecutive increases or decreases in the three subintervals.

More moderate, or midrange, results such as S can be arrived at by more than one path;

any combination of two price increases and one decrease will result in stock price S . Thus,

the midrange values will be more likely. The probability of each outcome is described by the

binomial distribution, and this multiperiod approach to option pricing is called the binomial binomial model

model. An option valuation

For example, using our initial stock price of $100, equal probability of stock price increases model predicated on

or decreases, and three intervals for which the possible price increase is 5% and the decrease the assumption that

stock prices can

is 3%, we can obtain the probability distribution of stock prices from the following calcula-

move to only two

tions. There are eight possible combinations for the stock price movement in the three periods:

values over any

. Each has a probability of 1„8. Therefore, the prob-

, , , , , , , short time period.

ability distribution of stock prices at the end of the last interval would be as follows.

Event Probability Stock Price

1.053

1„8

3 up movements $100 $115.76

1.052

2 up and 1 down 3„8 $100 0.97 $106.94

0.972

1 up and 2 down 3„8 $100 1.05 $ 98.79

0.973

3 down movements 1„8 $100 $ 91.27

The midrange values are three times as likely to occur as the extreme values. Figure 15.2A

is a graph of the frequency distribution for this example. The graph begins to exhibit the ap-

pearance of the familiar bell-shaped curve. In fact, as the number of intervals increases, as in

Figure 15.2B, the frequency distribution progressively approaches the lognormal distribution

rather than the normal distribution.2

Suppose we were to continue subdividing the interval in which stock prices are posited to

move up or down. Eventually, each node of the event tree would correspond to an infini-

tesimally small time interval. The possible stock price movement within that time interval

would be correspondingly small. As those many intervals passed, the end-of-period stock

2

Actually, more complex considerations enter here. The limit of this process is lognormal only if we assume also that

stock prices move continuously, by which we mean that over small time intervals only small price movements can

occur. This rules out rare events such as sudden, extreme price moves in response to dramatic information (like

a takeover attempt). For a treatment of this type of “jump process,” see John C. Cox and Stephen A. Ross, “The

Valuation of Options for Alternative Stochastic Processes,” Journal of Financial Economics 3 (January“March 1976),

pp. 145“66; or Robert C. Merton, “Option Pricing When Underlying Stock Returns Are Discontinuous,” Journal of

Financial Economics 3 (January“March 1976), pp. 125“44.

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

Essentials of Investments, Companies, 2003

Fifth Edition

540 Part FIVE Derivative Markets

F I G U R E 15.2 Probability

Probability distributions

A. Possible outcomes

and associated probabilities A 3/8

for stock prices after three

1/4

periods. The stock price starts

1/8

at $100, and in each period

it can increase by 5% or Future stock price

decrease by 3%.

90 100 110 115

B. Each period is subdivided Probability

into two smaller subperiods.

Now there are six periods,

and in each of these the stock

price can increase by 2.5% or

B 3/8

fall by 1.5%. As the number

1/4

of periods increases, the

stock price distribution 1/8

approaches the familiar Future stock price

bell-shaped curve.

90 100 110 115

price would more and more closely resemble a lognormal distribution. Thus, the apparent

oversimplification of the two-state model can be overcome by progressively subdividing any

period into many subperiods.

At any node, one still could set up a portfolio that would be perfectly hedged over the next

tiny time interval. Then, at the end of that interval, on reaching the next node, a new hedge

ratio could be computed and the portfolio composition could be revised to remain hedged over

the coming small interval. By continuously revising the hedge position, the portfolio would

remain hedged and would earn a riskless rate of return over each interval. This is called dy-

namic hedging, the continued updating of the hedge ratio as time passes. As the dynamic

hedge becomes ever finer, the resulting option valuation procedure becomes more precise.

>

5. Would you expect the hedge ratio to be higher or lower when the call option is

Concept

more in the money?

CHECK

15.3 BL ACK-SCHOLES OPTION VALUATION

While the binomial model we have described is extremely flexible, it requires a computer to

be useful in actual trading. An option-pricing formula would be far easier to use than the

tedious algorithm involved in the binomial model. It turns out that such a formula can be

derived if one is willing to make just two more assumptions: that both the risk-free interest

rate and stock price volatility are constant over the life of the option.

Black-Scholes

pricing formula The Black-Scholes Formula

A formula to value an

Financial economists searched for years for a workable option-pricing model before Black

option that uses the

and Scholes (1973) and Merton (1973) derived a formula for the value of a call option.

stock price, the risk-

Now widely used by options market participants, the Black-Scholes pricing formula for a

free interest rate, the

time to maturity, and European-style call option is

the standard deviation

T rT

C0 S0e N(d1) Xe N(d2) (15.1)

of the stock return.

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

Essentials of Investments, Companies, 2003

Fifth Edition

541

15 Option Valuation

F I G U R E 15.3

N(d) = Shaded area A standard

normal curve

d

0

where

2

ln(S0 /X) (r /2)T

d1

T

d2 d1 T

and where

C0 Current call option value.

S0 Current stock price.

N(d) The probability that a random draw from a standard normal distribution will

be less than d. This equals the area under the normal curve up to d, as in

the shaded area of Figure 15.3.

X Exercise price.

e 2.71828, the base of the natural log function.

Annual dividend yield of underlying stock. (We assume for simplicity that the

stock pays a continuous income flow, rather than discrete periodic payments,

such as quarterly dividends.)

r Risk-free interest rate, expressed as a decimal (the annualized continuously

compounded rate on a safe asset with the same maturity as the expiration date

of the option, which is to be distinguished from rf , the discrete period interest

rate).

T Time remaining until maturity of option (in years).

ln Natural logarithm function.

Standard deviation of the annualized continuously compounded rate of return

of the stock, expressed as a decimal, not a percent.

The option value does not depend on the expected rate of return on the stock. In a sense,

this information is already built into the formula with inclusion of the stock price, which itself

depends on the stock™s risk and return characteristics. This version of the Black-Scholes

formula is predicated on the assumption that the underlying asset has a constant dividend (or

income) yield.

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

Essentials of Investments, Companies, 2003

Fifth Edition

542 Part FIVE Derivative Markets

Although you may find the Black-Scholes formula intimidating, we can explain it at a

somewhat intuitive level. Consider a nondividend paying stock, for which 0. Then

T

S0e S0.

The trick is to view the N(d ) terms (loosely) as risk-adjusted probabilities that the call op-

tion will expire in the money. First, look at Equation 15.1 assuming both N(d ) terms are close

to 1.0; that is, when there is a very high probability that the option will be exercised. Then the

call option value is equal to S0 Xe rT, which is what we called earlier the adjusted intrinsic

value, S0 PV(X). This makes sense; if exercise is certain, we have a claim on a stock with

current value S0 and an obligation with present value PV(X), or with continuous compound-

ing, Xe rT.

Now look at Equation 15.1, assuming the N(d ) terms are close to zero, meaning the option

almost certainly will not be exercised. Then the equation confirms that the call is worth noth-

ing. For middle-range values of N(d ) between 0 and 1, Equation 15.1 tells us that the call

value can be viewed as the present value of the call™s potential payoff adjusting for the proba-

bility of in-the-money expiration.

How do the N(d ) terms serve as risk-adjusted probabilities? This question quickly leads us

into advanced statistics. Notice, however, that d1 and d2 both increase as the stock price in-

creases. Therefore, N(d1) and N(d2) also increase with higher stock prices. This is the property

we would desire of our “probabilities.” For higher stock prices relative to exercise prices,

future exercise is more likely.

You can use the Black-Scholes formula fairly easily. Suppose you want to value a call option

under the following circumstances:

15.2 EXAMPLE S0

Stock price 100

Black-Scholes X

Exercise price 95

Call Option