Valuation is a major theme in finance and investments. The valuation of bonds, preferred stock, and common stock composes a substantial proportion of the chapters devoted to these securities. The valuation of options is also important but is more difficult than most of the material covered in this text. This section will briefly cover the model initially devel- oped by Fischer Black and Myron Scholes for the valuation of warrants and subsequently applied to call options.1 This valuation model, commonly referred to as Black-Scholes, per- meates the literature on put and call options. It has also been applied to other areas of fi- nance in which there are options. For example, if a firm has the right to retire a bond issue prior to maturity, the bond has a built-in option. By valuing the option and separating that value from the amount of the debt, the financial analyst determines the cost of the debt.
The following discussion explains and illustrates the Black-Scholes option valua- tion model. (The derivation of the model is not given, so you will have to take the model on faith.) The question of valuation of an option is illustrated in Figure 18.1, which essentially reproduces Figure 17.2. Lines AB and BC represent the option’s intrinsic value, and line DE represents all the values of the option to buy for the various prices of the stock. The questions are: “Why is line DE located where it is? Why isn’t line DE higher or lower in the plane? What variables cause the line to shift up or down?” The Black-Scholes model determines the value of the option for each price of the stock and thus locates DE in the plane.
In Black-Scholes, the value of a call option (Vo) depends on all of the following:
Ps, the current price of the stock Pe, the option’s strike price
fIGuRE 18.1
Profits and Losses at Expiration for the Buyer of a Call
Price of the Stock
$50 C
A B
Price of the Option
0 50 $100
D
E
1The initial model was published in Fischer Black and Myron Scholes, “The Pricing of Options and Corporate Liabilities,”
Journal of Political Economy (May/June 1973): 637–654.
Source: © Cengage Learning
T, the time in years to the option’s expiration date (i.e., if expiration is 3 months, T 5 0.25)
s, the standard deviation of the stock’s annual rate of return
r, the annual risk-free rate of interest on an asset (e.g., Treasury bill) with a term equal to the time to the option’s expiration
The relationships between the value of a call (the dependent variable) and each of these independent variables (assuming the remaining variables are held constant) are as follows:
• An increase in the price of the stock (an increase in Ps) increases the value of a call option (Vo). This is true since the intrinsic value of the option rises as the price of the stock rises.
• An increase in the strike price (an increase in Pe ) decreases the value of a call option. Higher strike prices reduce the option’s intrinsic value for a given price of the stock.
• An increase in the time to expiration (an increase in T) increases the value of a call option. As time diminishes and the option approaches expiration, its value declines.
• An increase in the variability of the stock (an increase in s) increases the value of a call option. A speculator will find an option on a volatile stock more attractive than an option on a stock whose price tends to be stable. Decreased variability decreases the value of an option.
• An increase in interest rates (an increase in r) increases the value of a call option.
Higher interest rates are associated with higher call option valuations.
Most of the relationships between the independent variables and an option’s value seem reasonable with the exception of a change in the interest rate. Throughout this text, an increase in interest rates decreases the value of the asset. Higher interest rates reduce the present value of a bond’s interest payments and principal repayment, thus reducing the value of the bond. Higher interest rates increase the required return for a common stock, thus decreasing the valuation of the common stock. This negative rela- tionship between changes in interest rates and a security’s value does not hold for call options. Higher interest rates increase the value of an option to buy stock.
Although the positive relationship between interest rates and the value of a call op- tion seems perverse, the relationship makes sense. Remember that the intrinsic value of a call option is the difference between the price of the stock and the strike price. The inves- tor, however, does not have to exercise the call option immediately but may wait until its expiration. The funds necessary to exercise the option may be invested elsewhere. Higher interest rates mean these funds earn more. You need to invest less at the higher rate to have the funds to exercise the option at expiration. Thus the present value of the strike price (i.e., the funds necessary to exercise the call option) declines as interest rates rise.
This reduction in the present value of the strike price increases the value of the option.
It should be noted that dividends are excluded from the Black-Scholes model. In its initial formulation, the valuation model was applied to options on stocks that did not pay a dividend. Hence the dividend played no role in the determination of the option’s value. The model has been extended to dividend-paying stocks. Since the extension does not significantly change the basic model, this discussion will be limited to the original presentation.
Black-Scholes puts the variables together in the following equation for the value of a call option (Vo):
Vo5Ps3F1d12 2 Pe
erT 3F1d22. 18.1
The value of a call depends on two pieces: the price of the stock times a function, F(d1);
and the strike price, expressed in present value terms, times a function, F(d2). While the price of the stock (Ps) presents no problem, the strike price (Pe) expressed as a present value (Pe/erT) needs explanation. The strike price is divided by the number e 5 2.71828 raised to rT, the product of the risk-free interest rate and the option’s time to expira- tion. The use of e 5 2.71828 expresses compounding on a continuous basis instead of discrete (e.g., quarterly or monthly) time periods.
The definitions of the functions F(d1) and F(d2) are
d15 lnaPs
Peb 1 ar1s2 2bT
s"T 18.2
and
d25d12 s"T. 18.3
The ratio of the price of the stock and the strike price (Ps/Pe) is expressed as a natural logarithm (ln). The numerical values of d1 and d2 represent the area under the normal probability distribution. Applying Black-Scholes requires a table of the values for the cumulative normal probability distribution. Such a table is readily available in statistics textbooks, and one is provided in Exhibit 18.1 for convenience. Once d1 and d2 have been determined and the values from the cumulative probability distribution located, it is these values that are used in the Black-Scholes model (i.e., substituted for F(d1) and F(d2) in Equation 18.1).
How the model is applied may be seen by the following example. The values of the variables are
Stock price (Ps) $52
Strike price (Pe) $50
Time to expiration (T) 0.25 (three months)
Standard deviation (s) 0.20
Interest rate (r) 0.10 (10% annually)
Thus the values of d1 and d2 are
d15 lna52
50b 1 a0.11 0.22
2 b 30.25 0.2"0.25
5 0.03921 10.110.0220.25
0.1 50.692
ExhIBIT 1 8 .1 Cumulative Normal Distribution dF(d)dF(d)dF(d)dF(d)dF(d)dF(d)dF(d)dF(d)dF(d)dF(d)dF(d) −3.090.001−2.510.0060−1.930.0268−1.350.0885−0.770.2207−0.190.42470.390.65170.940.82641.490.93192.040.97932.590.9952 −3.080.001−2.500.0062−1.920.0274−1.340.0901−0.760.2236−0.180.42860.400.65540.950.82891.50.93322.050.97982.60.9953 −3.070.0011−2.490.0064−1.910.0281−1.330.0918−0.750.2266−0.170.43250.410.65910.960.83151.510.93452.060.98032.610.9955 −3.060.0011−2.480.0066−1.900.0287−1.320.0934−0.740.2297−0.160.43640.420.66280.970.8341.520.93572.070.98082.620.9956 −3.050.0011−2.470.0068−1.890.0294−1.310.0951−0.730.2327−0.150.44040.430.66640.980.83651.530.9372.080.98122.630.9957 −3.040.0012−2.460.0069−1.880.0301−1.300.0968−0.720.2358−0.140.44430.440.670.990.83891.540.93822.090.98172.640.9959 −3.030.0012−2.450.0071−1.870.0307−1.290.0985−0.710.2389−0.130.44830.450.67361.000.84131.550.93942.10.98212.650.996 −3.020.0013−2.440.0073−1.860.0314−1.280.1003−0.700.242−0.120.45220.460.67721.010.84381.560.94062.110.98262.660.9961 −3.010.0013−2.430.0075−1.850.0322−1.270.102−0.690.2451−0.110.45620.470.68081.020.84611.570.94182.120.9832.670.9962 −3.000.0013−2.420.0078−1.840.0329−1.260.1038−0.680.2483−0.100.46020.480.68441.030.84851.580.94292.130.98342.680.9963 −2.990.0014−2.410.0080−1.830.0336−1.250.1057−0.670.2514−0.090.46410.490.68791.040.85081.590.94412.140.98382.690.9964 −2.980.0014−2.400.0082−1.820.0344−1.240.1075−0.660.2546−0.080.46810.500.69151.050.85311.60.94522.150.98422.70.9965 −2.970.0015−2.390.0084−1.810.0351−1.230.1093−0.650.2578−0.070.47210.510.6951.060.85541.610.94632.160.98462.710.9966 −2.960.0015−2.380.0087−1.800.0359−1.220.1112−0.640.2611−0.060.47610.520.69851.070.85771.620.94742.170.9852.720.9967 −2.950.0016−2.370.0089−1.790.0367−1.210.1131−0.630.2643−0.050.48010.530.70191.080.85991.630.94842.180.98542.730.9968 −2.940.0016−2.360.0091−1.780.0375−1.200.1151−0.620.2676−0.040.4840.540.70541.090.86211.640.94952.190.98572.740.9969 −2.930.0017−2.350.0094−1.770.0384−1.190.117−0.610.2709−0.030.4880.550.70881.100.86431.650.95052.20.98612.750.997 −2.920.0018−2.340.0096−1.760.0392−1.180.119−0.600.2743−0.020.4920.560.71231.110.86651.660.95152.210.98642.760.9971 −2.910.0018−2.330.0099−1.750.0401−1.170.121−0.590.2776−0.010.4960.570.71571.120.86861.670.95252.220.98682.770.9972 −2.900.0019−2.320.0102−1.740.0409−1.160.123−0.580.2810.000.50.580.7191.130.87081.680.95352.230.98712.780.9973 −2.890.0019−2.310.0104−1.730.0418−1.150.1251−0.570.28430.010.504d2→0.590.72241.140.87291.690.95452.240.98752.790.9974 −2.880.002−2.300.0107−1.720.0427−1.140.1271−0.560.28770.020.5080.600.72571.150.87491.70.95542.250.98782.80.9974 −2.870.0021−2.290.0110−1.710.0436−1.130.1292−0.550.29120.030.5120.610.72911.160.8771.710.95642.260.98812.810.9975 −2.860.0021−2.280.0113−1.700.0446−1.120.1314−0.540.29460.040.5160.620.73241.170.8791.720.95732.270.98842.820.9976 −2.850.0022−2.270.0116−1.690.0455−1.110.1335−0.530.29810.050.51990.630.73571.180.8811.730.95822.280.98872.830.9977 −2.840.0023−2.260.0119−1.680.0465−1.100.1357−0.520.30150.060.52390.640.73891.190.8831.740.95912.290.9892.840.9977 −2.830.0023−2.250.0122−1.670.0475−1.090.1379−0.510.3050.070.52790.650.74221.200.88491.750.95992.30.98932.850.9978 −2.820.0024−2.240.0125−1.660.0485−1.080.1401−0.500.30850.080.53190.660.74541.210.88691.760.96082.310.98962.860.9979 −2.810.0025−2.230.0129−1.650.0495−1.070.1423−0.490.31210.090.53590.670.74861.220.88881.770.96162.320.98982.870.9979 −2.800.0026−2.220.0132−1.640.0505−1.060.1446−0.480.31560.10.53980.680.75171.230.89071.780.96252.330.99012.880.998 −2.790.0026−2.210.0136−1.630.0516−1.050.1469−0.470.31920.110.5438 d2→0.690.75491.240.89251.790.96332.340.99042.890.9981 −2.780.0027−2.200.0139−1.620.0526−1.040.1492−0.460.32280.120.54780.70.7581.250.89431.80.96412.350.99062.90.9981
dF(d)dF(d)dF(d)dF(d)dF(d)dF(d)dF(d)dF(d)dF(d)dF(d)dF(d) −2.770.0028−2.190.0143−1.610.0537−1.030.1515−0.450.32640.130.55170.710.76111.260.89621.810.96492.360.99092.910.9982 −2.760.0029−2.180.0146−1.600.0548−1.020.1539−0.440.330.140.55570.720.76421.270.8981.820.96562.370.99112.920.9982 −2.750.003−2.170.0150−1.590.0559−1.010.1562−0.430.33360.150.55960.730.76731.280.89971.830.96642.380.99132.930.9983 −2.740.0031−2.160.0154−1.580.0571−1.000.1587−0.420.33720.160.56360.740.77031.290.90151.840.96712.390.99162.940.9984 −2.730.0032−2.150.0158−1.570.0582−0.990.1611−0.410.34090.170.56750.750.77341.300.90321.850.96782.40.99182.950.9984 −2.720.0033−2.140.0162−1.560.0594−0.980.1635− 0.400.34460.180.57140.760.77641.310.90491.860.96862.410.9922.960.9985 −2.710.0034−2.130.0166−1.550.0606−0.970.166−0.390.34830.190.57530.770.77931.320.90661.870.96932.420.99222.970.9985 −2.70.0035−2.120.0170−1.540.0618−0.960.1685−0.380.3520.20.57930.780.78231.330.90821.880.96992.430.99252.980.9986 −2.690.0036−2.110.0174−1.530.063−0.950.1711−0.370.35570.210.58320.790.78521.340.90991.890.97062.440.99272.990.9986 −2.680.0037−2.100.0179−1.520.0643−0.940.1736−0.360.35940.220.58710.800.78811.350.91151.90.97132.450.992930.9987 −2.670.0038−2.090.0183−1.510.0655−0.930.1762−0.350.36320.230.5910.810.7911.360.91311.910.97192.460.99313.010.9987 −2.660.0039−2.080.0188− 1.500.0668−0.920.1788−0.340.36690.24.0.59480.820.79391.370.91471.920.97262.470.99323.020.9987 −2.650.004−2.070.0192−1.490.0681−0.910.1814−0.330.37070.250.59870.830.79671.380.91621.930.97322.480.99343.030.9988 −2.640.0041−2.060.0197−1.480.0694−0.900.1841−0.320.37450.260.60260.840.79951.390.91771.940.97382.490.99363.040.9988 −2.630.0043−2.050.0202−1.470.0708−0.890.1867−0.310.37830.270.60640.850.80231.400.91921.950.97442.50.99383.050.9989 −2.620.0044−2.040.0207−1.460.0721−0.880.1894−0.880.38210.280.61030.860.80511.410.92071.960.9752.510.9943.060.9989 −2.610.0045−2.030.0212−1.450.0735−0.870.1922−0.290.38590.290.61410.870.80781.420.92221.970.97562.520.99413.070.9989 −2.600.0047−2.020.0217−1.440.0749−0.860.1949−0.280.38970.30.61790.880.81061.430.92361.980.97612.530.99433.080.999 −2.590.0048−2.010.0222−1.430.0764−0.850.1977−0.270.39360.310.62170.890.81331.440.92511.990.97672.540.99453.090.999 −2.580.0049− 2.000.0228−1.420.0778−0.840.2005−0.260.39740.320.62550.900.81591.450.926520.97722.550.9946 −2.570.0051−1.990.0233−1.410.0793−0.830.2033−0.250.40130.330.62930.910.81861.460.92792.010.97782.560.9948 −2.560.0052−1.980.0239−1.400.0808−0.820.2061−0.240.40520.340.63310.920.82121.470.92922.020.97832.570.9949 −2.550.0054−1.970.0244−1.390.0823−0.810.209−0.230.4090.350.63680.930.82381.480.93062.030.97882.580.9951 −2.540.0055−1.960.0250−1.380.0838−0.800.2119−0.220.41290.360.6406 −2.530.0057−1.950.0256−1.370.0853−0.790.2148−0.210.41680.370.6443 −2.520.0059−1.940.0262−1.360.0869−0.780.2177 −0.200.42070.380.648 Significance LevelCritical Values of z for Two TailsLower TailUpper Tail 0.10±1.65−1.28+1.28 0.05±1.96−1.65+1.65 0.01±2.58−2.33+2.33
Exhibit 18.1 (Continued)
and
d250.69220.2"0.2550.69220.150.592.
The values from the normal distribution are2 F(0.692) ≈ 0.755 F(0.592) ≈ 0.722.
These values are represented by d1 and d2 in Figure 18.2, which shows the areas under the normal probability distribution for both d1 and d2. (The total shaded area repre- sents d1, while the checkerboard area represents d2.)
The probability distribution seeks to measure the probability of the option being exercised. If there is a large probability that the option will have positive intrinsic value at expiration, the numerical values of Fd1 and Fd2 approach 1, and the option’s value will approach the price of the stock minus the present value of the strike price:
Vo5 1Ps2 112 2 Pe
erT1125 1Ps2 2 Pe
erT.
If there is little probability that the option will have positive intrinsic value at expira- tion, the numerical values of d1 and d2 will approach 0, and the option will have little value:
Vo5 1Ps2 102 2 1Pe2
erT 102 50.
fIGuRE 18.2
A Normal Curve with the Areas for d1 and d2
0.5 0.722 0.755
2F(0.69) 5 0.7549 and F(0.59) 5 0.7224, which approximates the values given in the text.
Source: © Cengage Learning
Given the values for F(d1) and F(d2) determined from the normal distribution, the value of the call option is
Vo5 1$522 10.75522 $50
2.7182810.1210.25210.7222 5$4.00.
If the call is selling for more than $4.00, it is overvalued. If it is selling for less, it is undervalued.
If the price of the stock had been $60, the Black-Scholes model determines the value of the option to be $11.25. If the price of the stock were $40, the value of the option is $0.04. By altering the price of the stock, the various values of the option are determined. As shown in Figure 18.3, the different prices of the stock generate the gen- eral pattern of option values illustrated by line DE in Figure 18.1.
If one of the other variables (i.e., T, s, Pe, and r) were to change while holding the price of the stock constant, the curve representing the value of the option would shift.
If the life of the option had been nine months instead of three months, the curve would shift up. Increased price volatility, a lower strike price, or higher interest rates would also shift the Black-Scholes option valuation curve upwards. A shorter time to expira- tion, a lower interest rate, a higher strike price, or smaller volatility would shift the curve downward.
These relationships are illustrated in Exhibit 18.2, which shows the impact of each variable on a call’s value using Black-Scholes. This illustration uses the previous example and is divided into five cases. In each case one of the variables is changed while all the others are held constant. The value derived in the initial illustration is underlined in each case. In case 1, the price of the stock varies from $40 to $70, and as the price of the stock rises, so does the valuation of the option. When the option is way out of the money (i.e., when the stock is selling for $40), the valuation is a minimal $0.04. The value rises to
fIGuRE 18.3
Black-Scholes Call Option Values
Price of the Stock
$11.25
C
A
B Price of
the Option
$40 60
D
E
50 4.00
0.04
52
Intrinsic Value Black-Scholes
Call Option Values
Source: © Cengage Learning
w
ExhIBIT 18.2
Black-Scholes Option Valuations Initial values:
Price of the stock $52.00
Strike price $50.00
Time to expiration 0.25 (three months, or 90 days)
Standard deviation 0.20
Risk-free interest rate 0.10 (10 percent annually)
Black-Scholes valuation $ 4.00
Case 1: Price of the Stock Is Altered Case 2: Strike Price Is Altered
Stock Price
Black-Scholes
Option Value Strike Price
Black-Scholes Option Value
$40 $0.04 $40 $12.98
45 0.55 45 8.18
50 2.62 50 4.00
52 4.00 55 1.37
55 6.50 60 0.31
60 11.25 65 0.03
65 16.22 70 0.01
70 21.22
Case 3: Time to Expiration Is Altered Case 4: Standard Deviation Is Altered
Days
Black-Scholes Option Value
Standard Deviation
Black-Scholes Option Value
360 $8.08 1.0 $11.56
270 6.86 0.6 7.71
180 5.53 0.3 4.87
90 4.00 0.2 4.00
60 3.41 0.15 3.62
30 2.74 0.1 3.34
15 2.36 0.05 3.22
7 2.14 0.001 3.21
1 2.01
Case 5: Interest Rate Is Altered
Interest Rate Black-Scholes Option Value
0.20 $4.91
0.15 4.45
0.12 4.18
0.10 4.00
0.08 3.83
0.06 3.66
0.04 3.50
In case 2, the strike price varies from $40 to $70. As would be expected, the value of the option declines with higher strike prices. Although the option is worth $12.98 when the strike price is $40, the option is virtually worthless at a strike price of $70.
Case 3 illustrates the decline in the value of the option as it approaches expiration.
A year prior to expiration, the option at $50 is worth $8.08 when the stock sells for $52.
This value declines to $4.00 when three months remain. With two weeks to expiration, the option is worth $2.36, and at expiration, the option is worth only its $2.00 intrinsic value.
In case 4, the variability of the underlying stock’s return is altered. Greater vari- ability usually decreases the attractiveness of a security, but the opposite occurs with call options. Increased variability means there is a greater chance the underlying stock’s price will rise and increase the intrinsic value of the option. Thus, increased variability is associated with higher option valuations, and lower variability is associated with lower option valuations. This relationship is seen in case 4. As the standard deviation of the stock’s return declines, so does the value of the option.
In the last case, the interest rate is changed. As was explained earlier, a higher inter- est rate decreases the present value of the strike price and increases the value of the call option. This relationship is seen in case 5. At an annual interest rate of 20 percent, the option is worth $4.91, but this value decreases as the interest rate declines.
Although Black-Scholes may appear formidable, it is easily applied because computer programs have been developed to perform the calculations. All the variables but one are readily observable. Unfortunately, the standard deviation of the stock’s return is not observ- able, so the individual will have to develop a means to obtain that data to apply the model.
One method to overcome that problem is to reverse the equation and solve for the standard deviation. If the individual knows the price of the stock, the strike price, the price of the option, the term of the option, and the interest rate, Black-Scholes may be used to solve for the standard deviation of the returns. Historical data are then used in Black-Scholes to determine the implied historical variability of the underlying stock’s returns. If it can be assumed that the variability has not changed, then that value for the standard deviation is assumed to be the correct measure of the stock’s current vari- ability and is used to determine the present value of an option.