Interest rate 0.10

Valuation

Dividend yield 0

T

Time to expiration 0.25 (one-quarter year)

Standard deviation 0.50

First calculate

0.52/2)0.25

ln(100/95) (0.10 0

d1 0.43

0.5 0.25

d2 0.43 0.5 0.25 0.18

Next find N(d1) and N(d2). The normal distribution function is tabulated and may be found

in many statistics textbooks. A table of N(d) is provided on page 544 as Table 15.2. The nor-

mal distribution function N(d), is also provided in any spreadsheet program. In Microsoft

Excel, for example, the function name is NORMSDIST. Using either Excel or Table 15.2

(using interpolation for 0.43), we find that

N(0.43) 0.6664

N(0.18) 0.5714

0, S0e S0. Thus, the value of the call option is

T

Finally, remember that with

C 0.10 0.25

100 0.6664 95e 0.5714

66.64 52.94 $13.70

>

6. Calculate the call option value if the standard deviation on the stock is 0.6 instead

Concept

of 0.5. Confirm that the option is worth more using this higher volatility.

CHECK

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

Essentials of Investments, Companies, 2003

Fifth Edition

543

15 Option Valuation

What if the option price in Example 15.2 were $15 rather than $13.70? Is the option mis-

priced? Maybe, but before betting your career on that, you may want to reconsider the valua-

tion analysis. First, like all models, the Black-Scholes formula is based on some simplifying

abstractions that make the formula only approximately valid.

Some of the important assumptions underlying the formula are the following:

1. The stock will pay a constant, continuous dividend yield until the option expiration date.

2. Both the interest rate, r, and variance rate, 2, of the stock are constant (or in slightly

more general versions of the formula, both are known functions of time”any changes are

perfectly predictable).

3. Stock prices are continuous, meaning that sudden extreme jumps, such as those in the

aftermath of an announcement of a takeover attempt, are ruled out.

Variants of the Black-Scholes formula have been developed to deal with some of these

limitations.

Second, even within the context of the Black-Scholes model, you must be sure of the ac-

curacy of the parameters used in the formula. Four of these”S0, X, T, and r”are straight-

forward. The stock price, exercise price, and time to maturity are readily determined. The

interest rate used is the money market rate for a maturity equal to that of the option, and the

dividend yield is usually reasonably stable, at least over short horizons.

The last input, though, the standard deviation of the stock return, is not directly observable.

It must be estimated from historical data, from scenario analysis, or from the prices of other

options, as we will describe momentarily. Because the standard deviation must be estimated,

it is always possible that discrepancies between an option price and its Black-Scholes value

are simply artifacts of error in the estimation of the stock™s volatility. implied volatility

In fact, market participants often give the option valuation problem a different twist. Rather

The standard

than calculating a Black-Scholes option value for a given stock standard deviation, they ask

deviation of stock

instead: What standard deviation would be necessary for the option price that I can observe to returns that is

be consistent with the Black-Scholes formula? This is called the implied volatility of the op- consistent with an

tion, the volatility level for the stock that the option price implies. Investors can then judge option™s market value.

WEBMA STER

E-Investments: Black-Scholes Option Pricing

Go to options calculator available at www.schaefferresearch.com/stock/calculator.asp.

Use EMC Corporation for the firm. Enter the ticker symbol (EMC) and latest price for

the firm. Since the company is not paying a cash dividend at this time, enter 0.0 for the

quarterly dividend. The calculator will display the current interest rate. Find the prices for

call and put options in the two months following the closest expiration month. (You can

request the options prices directly in the calculator.) For example, if you are in February,

you would use the April and July options. Use the options that are closest to being at

the money. For example, if the most recent price of EMC was $56.40, you would select the

55 strike price.

Once you have entered the options prices and other data, hit the Go Figure button and

analyze the results.

1. Are the calculated prices in line with observed prices?

2. Compare the implied volatility with the historical volatility.

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

Essentials of Investments, Companies, 2003

Fifth Edition

544 Part FIVE Derivative Markets

d N(d ) d N(d ) d N(d )

TA B L E 15.2

3.00 0.0013 1.58 0.0571 0.76 0.2236

Cumulative

2.95 0.0016 1.56 0.0594 0.74 0.2297

normal

distribution 2.90 0.0019 1.54 0.0618 0.72 0.2358

2.85 0.0022 1.52 0.0643 0.70 0.2420

2.80 0.0026 1.50 0.0668 0.68 0.2483

2.75 0.0030 1.48 0.0694 0.66 0.2546

2.70 0.0035 1.46 0.0721 0.64 0.2611

2.65 0.0040 1.44 0.0749 0.62 0.2676

2.60 0.0047 1.42 0.0778 0.60 0.2743

2.55 0.0054 1.40 0.0808 0.58 0.2810

2.50 0.0062 1.38 0.0838 0.56 0.2877

2.45 0.0071 1.36 0.0869 0.54 0.2946

2.40 0.0082 1.34 0.0901 0.52 0.3015

2.35 0.0094 1.32 0.0934 0.50 0.3085

2.30 0.0107 1.30 0.0968 0.48 0.3156

2.25 0.0122 1.28 0.1003 0.46 0.3228

2.20 0.0139 1.26 0.1038 0.44 0.3300

2.15 0.0158 1.24 0.1075 0.42 0.3373

2.10 0.0179 1.22 0.1112 0.40 0.3446

2.05 0.0202 1.20 0.1151 0.38 0.3520

2.00 0.0228 1.18 0.1190 0.36 0.3594

1.98 0.0239 1.16 0.1230 0.34 0.3669

1.96 0.0250 1.14 0.1271 0.32 0.3745

1.94 0.0262 1.12 0.1314 0.30 0.3821

1.92 0.0274 1.10 0.1357 0.28 0.3897

1.90 0.0287 1.08 0.1401 0.26 0.3974

1.88 0.0301 1.06 0.1446 0.24 0.4052

1.86 0.0314 1.04 0.1492 0.22 0.4129

1.84 0.0329 1.02 0.1539 0.20 0.4207

1.82 0.0344 1.00 0.1587 0.18 0.4286

1.80 0.0359 0.98 0.1635 0.16 0.4365

1.78 0.0375 0.96 0.1685 0.14 0.4443

1.76 0.0392 0.94 0.1736 0.12 0.4523

1.74 0.0409 0.92 0.1788 0.10 0.4602

1.72 0.0427 0.90 0.1841 0.08 0.4681

1.70 0.0446 0.88 0.1894 0.06 0.4761

1.68 0.0465 0.86 0.1949 0.04 0.4841

1.66 0.0485 0.84 0.2005 0.02 0.4920

1.64 0.0505 0.82 0.2061 0.00 0.5000

1.62 0.0526 0.80 0.2119 0.02 0.5080

1.60 0.0548 0.78 0.2177 0.04 0.5160

whether they think the actual stock standard deviation exceeds the implied volatility. If it does,

the option is considered a good buy; if actual volatility seems greater than the implied volatil-

ity, the option™s fair price would exceed the observed price.

Another variation is to compare two options on the same stock with equal expiration

dates but different exercise prices. The option with the higher implied volatility would be

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

Essentials of Investments, Companies, 2003

Fifth Edition

545

15 Option Valuation

d N(d ) d N(d ) d N(d )

TA B L E 15.2

0.06 0.5239 0.86 0.8051 1.66 0.9515

(concluded)

0.08 0.5319 0.88 0.8106 1.68 0.9535

0.10 0.5398 0.90 0.8159 1.70 0.9554

0.12 0.5478 0.92 0.8212 1.72 0.9573

0.14 0.5557 0.94 0.8264 1.74 0.9591

0.16 0.5636 0.96 0.8315 1.76 0.9608

0.18 0.5714 0.98 0.8365 1.78 0.9625

0.20 0.5793 1.00 0.8414 1.80 0.9641

0.22 0.5871 1.02 0.8461 1.82 0.9656

0.24 0.5948 1.04 0.8508 1.84 0.9671

0.26 0.6026 1.06 0.8554 1.86 0.9686

0.28 0.6103 1.08 0.8599 1.88 0.9699

0.30 0.6179 1.10 0.8643 1.90 0.9713

0.32 0.6255 1.12 0.8686 1.92 0.9726

0.34 0.6331 1.14 0.8729 1.94 0.9738

0.36 0.6406 1.16 0.8770 1.96 0.9750

0.38 0.6480 1.18 0.8810 1.98 0.9761

0.40 0.6554 1.20 0.8849 2.00 0.9772

0.42 0.6628 1.22 0.8888 2.05 0.9798

0.44 0.6700 1.24 0.8925 2.10 0.9821

0.46 0.6773 1.26 0.8962 2.15 0.9842

0.48 0.6844 1.28 0.8997 2.20 0.9861

0.50 0.6915 1.30 0.9032 2.25 0.9878

0.52 0.6985 1.32 0.9066 2.30 0.9893

0.54 0.7054 1.34 0.9099 2.35 0.9906

0.56 0.7123 1.36 0.9131 2.40 0.9918

0.58 0.7191 1.38 0.9162 2.45 0.9929

0.60 0.7258 1.40 0.9192 2.50 0.9938

0.62 0.7324 1.42 0.9222 2.55 0.9946

0.64 0.7389 1.44 0.9251 2.60 0.9953

0.66 0.7454 1.46 0.9279 2.65 0.9960

0.68 0.7518 1.48 0.9306 2.70 0.9965

0.70 0.7580 1.50 0.9332 2.75 0.9970

0.72 0.7642 1.52 0.9357 2.80 0.9974

0.74 0.7704 1.54 0.9382 2.85 0.9978

0.76 0.7764 1.56 0.9406 2.90 0.9981

0.78 0.7823 1.58 0.9429 2.95 0.9984

0.80 0.7882 1.60 0.9452 3.00 0.9986

0.82 0.7939 1.62 0.9474 3.05 0.9989

0.84 0.7996 1.64 0.9495

considered relatively expensive because a higher standard deviation is required to justify its

price. The analyst might consider buying the option with the lower implied volatility and writ-

ing the option with the higher implied volatility.

The Black-Scholes call-option valuation formula, as well as implied volatilities, are eas-

ily calculated using an Excel spreadsheet, as in Figure 15.4. The model inputs are listed in

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

Essentials of Investments, Companies, 2003

Fifth Edition

E XC E L Applications www.mhhe.com/bkm

> Black-Scholes Option Pricing

Figure 15.4 captures a portion of the Excel model “B-S Option.” The model is built to value puts

and calls and extends the discussion to include analysis of intrinsic value and time value of op-

tions. The spreadsheet contains sensitivity analyses on several key variables in the Black-Scholes

pricing model.

You can learn more about this spreadsheet model by using the interactive version available on

our website at www.mhhe.com/bkm.

F I G U R E 15.4

Spreadsheet to calculate Black-Scholes call-option values

column B, and the outputs are given in column E. The formulas for d1 and d2 are provided in

the spreadsheet, and the Excel formula NORMSDIST(d1) is used to calculate N(d1). Cell E6

contains the Black-Scholes call option formula. To compute an implied volatility, we can use

the Solver command from the Tools menu in Excel. Solver asks us to change the value of one

cell to make the value of another cell (called the target cell) equal to a specific value. For ex-

ample, if we observe a call option selling for $7 with other inputs as given in the spreadsheet,

we can use Solver to find the value for cell B2 (the standard deviation of the stock) that will

make the option value in cell E6 equal to $7. In this case, the target cell, E6, is the call price,

and the spreadsheet manipulates cell B2. When you ask the spreadsheet to “Solve,” it finds

that a standard deviation equal to .2783 is consistent with a call price of $7; therefore, 27.83%

would be the call™s implied volatility if it were selling at $7.

>

7. Consider the call option in Example 15.2 If it sells for $15 rather than the value of

Concept

$13.70 found in the example, is its implied volatility more or less than 0.5?

CHECK