104; 0.3/0.5
Using the polynomial approximation in Section 12.9 gives usN(d:) = 0.5800, —N(d) = 0.4959and, from equation (12.20), the call price is39.0259 x 0.5800 — 40¢7°*"> x 0.4959 = 3.67ot $3.67.
'4 In theory this is not quite the same as the volatility of the stochastic process followed by the whole stock price.The volatility of the risky component is approximately equal to the volatility of the whole stock price multiplied bySo/(So — D), where D is the present value of the dividends. However, an adjustment is only necessary whenvolatilities are estimated using historical data. An implied volatility is calculated after the present value of dividendshave been subtracted from the stock price and is the volatility of the risky component.254 CHAPTER 12
 
American Options
Consider next American call options. In Section 8.5, we showed that in the absence of dividendsAmerican options should never be exercised early. An extension to the argument shows that, whenthere are dividends, it is optimal to exercise only at a time immediately before the stock goes ex-dividend. We assume that n ex-dividend dates are anticipated and that t,, t2,..., f, are moments intime immediately prior to the stock going ex-dividend, with t; < t, < t,; <--- <t,. The dividendscorresponding to these times will be denoted by D,, D,..., D,, respectively.
We start by considering the possibility of early exercise just prior to the final ex-dividend date(i.e., at time ¢,). If the option is exercised at time t,, the investor receives
S(t,) — K
If the option is not exercised, the stock price drops to S$(t,) — D,. As shown by equation (8.5), thevalue of the option is then greater than
S(ty) — Dy — KeTt follows that ifS(t,) — D, — Ke" —™ > S(t,) — Kthat is,D, < K(1 — etn) (12.24)
it cannot be optimal to exercise at time t,. On the other hand, ifD, > KU -e-®) (12.25)
for any reasonable assumption about the stochastic process followed by the stock price, it can beshown that it is always optimal to exercise at time ¢, for a sufficiently high value of S(t,). Theinequality in (12.25) will tend to be satisfied when the final ex-dividend date is fairly close to thematurity of the option (i.e., 7 — f, is small) and the dividend is large.
Next consider time ¢,,_,, the penultimate ex-dividend date. If the option is exercised at time t,_1,
tee fapectar neceivesthe investor receives
S(n1) — K
If the option is not exercised at time t,_,, the stock price drops to S(t,_;) — D,_,; and the earliestsubsequent time at which exercise could take place is t,. Hence, from equation (8.5), a lower boundto the option price if it is not exercised at time t,_) is
Spa) — Dy = Kee?It follows that ifS(ta-1) ~ Di-1 ~ Kern) 2 S(tn-1) -KorD,-1 < KC = ena)
it is not optimal to exercise at time ¢,_,. Similarly, for any i < a, ifD; < K(L—e Ht) (12.26)
it is not optimal to exercise at time f;.The Black-Scholes Model 255
 
The inequality in (12.26) is approximately equivalent toD; < Kr(tiss — fj)
Assuming that K is fairly close to the current stock price, the dividend yield on the stock wouldhave to be either close to or above the risk-free rate of interest for this inequality not to be satisfied.This is not often the case.
We can conclude from this analysis that, in most circumstances, the only time that needs to beconsidered for the early exercise of an American call is the final ex-dividend date, 4. Furthermore,if inequality (12.26) holds for i= 1,2,...,2—1 and inequality (12.24) holds, then we can becertain that early exercise is never optimal.
Black’s Approximation
Black suggests an approximate procedure for taking account of early exercise in call options.’ Thisinvolves calculating, as described earlier in this section, the prices of European options that matureat times T and ¢,, and then setting the American price equal to the greater of the two. Thisapproximation seems to work well in most cases. A more exact procedure suggested by Roll (1977),Geske (1979), and Whaley (1981) is given in Appendix 12B.'¢
Example 12.9 Consider the situation in Example 12.8, but suppose that the option is Americanrather than European. In this case, D, = D; = 0.5, Sy = 40, K =40, r=0.09, tr) = 2/12, andty = 5/12. Because :
 
 
KL — e279) = 40(1 — 6 8) = 0.89
is greater than 0.5, it follows (see inequality (12.26)) that the option should never be exercisedimmediately before the first ex-dividend date. In addition, because
K(1 — eH) = 4o(y — 67 0.99*09833 — 9.30
is less than 0.5, it follows (see inequality (12.25)) that, when it is sufficiently deep in the money, theoption should be exercised immediately before the second ex-dividend date.We now use Black’s approximation to value the option. The present value of the first dividend is
0.5¢70:1667%0.09 _ 9 4996so that the value of the option, on the assumption that it expires just before the final ex-dividenddate, can be calculated using the Black-Scholes formula with Sj = 39.5074, K = 40, r= 0.09,o = 0.30, and T = 0.4167. It is $3.52. Black’s approximation involves taking the greater of thisand the value of the option when it can only be exercised at the end of six months. FromExample 12.8, we know that the latter is $3.67. Black’s approximation therefore gives the value ofthe American call as $3.67.
'S See F. Black, “Fact and Fantasy in the Use of Options,” Financial Analysts Journal, 31 (July/August 1975), 36-41,61-72.
'6 See R. Roll, “An Analytic Formuta for Unprotected American Call Options on Stocks with Known Dividends,”Journal of Financial Economics, 5 (1977), 25}-58; R. Geske, “A Note on an Analytic Valuation Formula forUnprotected American Caii Options on Stocks with Known Dividends,” Journal of Financial Economics, 7 (1979),375-80; R. Whaley, “On the Vatuation of American Calt Options on Stocks with Known Dividends,” Journal ofFinancial Economics, 9 (June 1981), 207-11; R. Geske, “Comments on Whaley’s Note,” Journal of FinancialEconomics, 9 (June 1981), 213-215.256 CHAPTER 12
 
The value of the option given by the Roll, Geske, and Whaley (RGW) formula is $3.72. There aretwo reasons for differences between RGW and Black’s approximation (BA). The first concerns thetiming of the early exercise decision and tends to make RGW greater than BA. In BA, theassumption is that the holder has to decide today whether the option will be exercised after fivemonths or after six months; RGW allows the decision on early exercise at the five-month point todepend on the stock price. The second concerns the way in which volatility is applied and tends tomake BA greater than RGW. In BA, when we assume exercise takes place after five months, thevolatility is applied to the stock price less the present value of the first dividend: when we assumeexercise takes place after six months, the volatility is applied to the stock price less the present valueof both dividends, In RGW, it is always applied to the stock price less the present value of bothdividends.
Whaley!’ has empirically tested three models for the pricing of American calls on dividend-paying stocks: (1) the formula in Appendix 12B; (2) Black’s model; and (3) the European optionpticing model described at the beginning of this section. He used {5,582 Chicago Board options.The models produced pricing errors with means of 1.08%, 1.48%, and 2.15%, respectively. Thetypical bid-offer spread on a call option is greater than 2.15% of the price. On average, therefore,all three models work well and within the tolerance imposed on the options market by tradingimperfections.
Up to now, our discussion has centered on American call options. The results for American putoptions are less clear-cut. Dividends make it less likely that an American put option will beexercised early. It can be shown that it is never worth exercising an American put for a periodimmediately prior to an ex-dividend date.’® Indeed, if
D> K- e itl)for alli <n andD, 2 K(1— 7”)
an argument analogous to that just given shows that the put option should never be exercised early.In other cases, numerical procedures must be used to value a put.
SUMMARY
 
 
We started this chapter by examining the properties of the process for stock prices introduced inChapter 11. The process implies that the price of a stock at some future time, given its price today,is lognormal. It also implies that the continuously compounded return from the stock in a periodof time is normally distributed. Our uncertainty about future stock prices increases as we lookfurther ahead. The standard deviation of the logarithm of the stock price is proportional to thesquare root of how far ahead we are looking.
To estimate the volatility, ¢, of a stock price empirically, the slock price is observed al fixedintervals of time\(e.g., every day, every week, or every month). For each time period, the naturallogarithm of te ati of the stock price at the end of the time period to the stock price at the
'7 See R. E. Whaley, “Valuation of American Calt Options on Dividend Paying Stocks: Empirical Tests,” Journal ofFinancial Economics, 10 (March 1982), 29-58.
'8 See H. EB. Johnson, “Three Topics in Option Pricing,’ Ph.D. thesis, University of California, Los Angeles, 1981,p. 42.The Black-Scholes Mode! 257
beginning of the time period is calculated. The volatility is estimated as the standard deviation ofthese numbers divided by the square root of the length of the time period in years. Usually, dayswhen the exchanges are closed are ignored in measuring time for the purposes of volatilitycalculations.
The differential equation for the price of any derivative dependent on a stock can be obtainedby creating a riskless position in the option and the stock. Because the derivative and the stockprice both depend on the same underlying source of uncertainty, this can always be done. Theposition that is created remains riskless for only a very short period of time. However, the returnon a tiskless position must always be the risk-free interest rate if there are to be no arbitrageopportunities.
The expected return on the stock does not enter into the Black-Scholes differential equation.This leads to a useful result known as risk-neutral valuation. This result states that when valuing aderivative dependent on a stock price, we can assume that the world is risk neutral. This meansthat we can assume that the expected return from the stock is the risk-free interest rate, and thendiscount expected payoffs at the risk-free interest rate. The Black-Scholes equations for Europeancall and put options can be derived by either solving their differential equation or by using risk-neutral valuation.
An implied volatility is the volatility that, when used in conjunction with the Black-Scholesoption pricing formula, gives the market price of the option. Traders monitor implied volatilitiesand commonly use the implied volatilities from actively traded options to estimate the appropriatevolatility to use to price a less actively traded option on the same asset. Empirical results show thatthe volatility of a stock is much higher when the exchange is open than when it is closed. Thissuggests that, to.some extent, trading itself causes stock price volatility.
The Black-Scholes results can be extended to cover European call and put options on dividend-
saving stocks. The nrocedure is to use the Black Scholes formula with the stock srice reduced bypaying StOCKS, anc proceaure is fo USC tac oIaCK SCHOICS TO(rMUa With tic siocK price requcca oY
the present value of the dividends anticipated during the life of the option, and the volatility equalto the volatility of the stock price net of the present value of these dividends.
In theory, American call options are liable to be exercised early, immediately before any ex-dividend date. In practice, only the final ex-dividend date usually needs to be considered. FischerBlack has suggested an approximation. This involves setting the American call option price equalto the greater of two European call option prices. The first European call option expires at thesame time as the American call option; the second expires immediately prior to the final ex-dividend date. A more exact approach involving bivariate normal distributions is explained inAppendix 12B.
SUGGESTIONS FOR FURTHER READING
 
 
On the Distribution of Stock Price Changes
Blattberg, R., and N. Gonedes, “A Comparison of the Stable and Student Distributions as StatisticalModels for Stock Prices,” Journal of Business, 47 (April 1974), 244-80.
Fama, E. F., “The Behavior of Stock Prices,” Journal of Business, 38 (January 1965), 34-105.Kon, S. J., “Models of Stock Returns—A Comparison,” Journal of Finance, 39 (March 1984), 147-65.
Richardson, M., and T. Smith, “A Test for Multivariate Normality in Stock Returns,” Journal ofBusiness, 66 (1993), 295-321.258 CHAPTER 12
On the Black-Scholes AnalysisBlack, F., and M. Scholes, ““The Pricing of Options and Corporate Liabilities,” Journal of PoliticalEconomy, 81 (May/June 1973), 637-59.
Merton, R. C., “Theory of Rational Option Pricing,” Belf Journal of Economics and ManagementScience, 4 (Spring 1973), 141-183.
On Risk-Neutral Valuation
Cox, J. C., and S. A. Ross, “The Valuation of Options for Alternative Stochastic Processes,” Journal ofFinancial Economics, 3 (1976), 145-66.
Smith, C. W., “Option Pricing: A Review,” Journal of Financial Economics, 3 (1976), 3-54.
Geske, R., “Comments on Whaley’s Note,” Journal of Financial Economics, 9 (June 1981), 213-15.
Geske, R., “A Note on an Analytic Valuation Formula for Unprotected American Call Options onStocks with Known Dividends,” Journal of Financial Economics, 7 (1979), 375-80.
Roll, R., “An Analytical Formula for Unprotected American Call Options on Stocks with KnownDividends,” Journal of Financial Economics, 5 (1977), 251-58.
Whaley, R., “On the Valuation of American Call Options on Stocks with Known Dividends,” Journalof Financial Economics, 9 (1981), 207-11.
On the Causes of Volatility
Fama, E. E., “The Behavior of Stock Market Prices,” Journal of Business, 38 (January 1965), 34-105.
French, K. R., “Stock Returns and the Weekend Effect,” Journal of Financial Economics, 8 (March 1980),55-69.
French, K. R., and R. Roll, “Stock Return Variances: The Arrival of Information and the Reaction ofTraders,” Journal of Financial Economics, 17 (September 1986), 5-26.
QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)
 
12.1. What does the Black-Scholes stock option pricing model assume about the probability
distribution of the stock srice in one vear? What docs it assume about the continucuslyaistrioution Of the SstocK price In one year!’ What cocs it assume avout inc continuousy
compounded rate of return on the stock during the year?
12.2. The volatility of a stock price is 30% per annum. What is the standard deviation of thepercentage price change in one trading day?
12.3. Explain the principle of risk-neutral valuation.
12.4. Calculate the price of a three-month European put option on a non-dividend-paying stock with astrike price of $50 when the current stock price is $50, the risk-free interest rate is 10% perannum, and the volatility is 30% per annum.
12.5. What difference does it make to your calculations in Problem 12.4 if a dividend of $1.50 isexpected in two months?The Black-Scholes Model 259
12.6.12.7.
12.8.
12.9.
12.10.
12.11.
12.12.
12.13.
12.14.
12.15.
What is implied volatility? How can it be calculated?
A stock price is currently $40. Assume that the expected return from the stock is 15% and thatits volatility is 25%. What is the probability distribution for the rate of return (with continuouscompounding) earned over a two-year period?
A stock price follows geometric Brownian motion with an expected return of 16% and avolatility of 35%. The current price is $38.a. What is the probability that a European call option on the stock with an exercise price of$40 and a maturity date in six months will be exercised?
b. What is the probability that a European put option on the stock with the same exerciseprice and maturity will be exercised?
Prove that, with the notation in the chapter, a 95% confidence interval for S; is betweenPp
2 2Syelt-# /2DT=1.960./T and Spel /2T+1.960/T
A portfolio manager announces that the average of the returns realized in each year of the last10 years is 20% per annum. In what respect is this statement misleading?
Assume that a non-dividend-paying stock has an expected return of x and a volatility of o.An innovative financial institution has just announced that it will trade a security that paysoff a dollar amount equal to In S; at time T, where S; denotes the value of the stock price attime T.a. Use risk-neutral valuation to calculate the price of the security at time ¢ in terms of thestock price, S, at time ¢.b. Confirm that your price satisfies the differential equation (12.15).
Consider a derivative that pays off S7 at time T, where Sz is the stock price at that time. Whenthe stock price follows geometric Brownian motion, it can be shown that its price at time ¢
(t < T) has the formA(t, T)S"
where S is the stock price at time ¢ and h is a function only of t and T.a. By substituting into the Black-Scholes partial differential equation, derive an ordinarydifferential equation satisfied by A(t, T).b. What is the boundary condition for the differential equation for h(t, T)?c. Show thatA(t, T) = gl0.5o?nfn—+ren— DT)
where r is the risk-free interest rate and o is the stock price volatility.
What is the price of a European call option on a non-dividend-paying stock when the stock priceis $52, the strike price is $50, the risk-free interest rate is 12% per annum, the volatility is 30% perannum, and the time to maturity is three months?
What is the price of a European put option on a non-dividend-paying stock when the stock priceis $69, the strike price is $70, the risk-free interest rate is 5% per annum, the volatility is 35% perannum, and the time to maturity is six months?
Consider an American call option on a stock. The stock price is $70, the time to maturity is eightmonths, the risk-free rate of interest is 10% per annum, the exercise price is $65, and thevolatility is 32%. A dividend of $1 is expected after three months and again after six months.Show that it can never be optimal to exercise the option on either of the two dividend dates. UseDerivaGem to calculate the price of the option.260
12.16.
12.17.
12.18.12.19.
12.20.
12.21.
12.22.
CHAPTER 12
A call option on a non-dividend-paying stock has a market price of $2}. The stock price is $15,the exercise price is $13, the time to maturity is three months, and the risk-free interest rate is 5%per annum. What is the implied volatility?With the notation used in this chapter
a. What is N’(x)?
b. Show that SN’(d,) = Ke~"7-? N'(d,), where S is the stock price at time ¢ and
_ In(S/K) + (¢ +.0°/2KT — 1)oJ/T—t
_ In(S/K) +r — 0° /2KT — 1)
~ aVJ/T —t
ady
. Calculate dd, /dS and dd,/dS. /. Show that, when ¢ = SN(d\)— Ke’ 9 N(d)),dc o— = —rKe™ 9 N(d,) — SN'(d,) ———a rKe (a2) (di) D/Fotwhere c is the price of a call option on a non-dividend-paying stock.. Show that dc/aS = N(d).Show that c satisfies the Black-Scholes differential equation.. Show that c satisfics the boundary condition for a European call option, i.e., thatc=max(S — K,0) ast T.
an
gem oO
Show that the Black-Scholes formulas for call and put options satisfy put—call parity.
A stock price is currently $50 and the risk-free interest rate is 5%. Use the DerivaGem softwareto translate the following table of European call options on the stack into a table of implied
Transiate the followin Buropean call ons on thi Ock into a table of imp
volatilities, assuming no dividends:
 
Maturity (months)
 
Strike price ($) 3 6 1245 7.0 8.3 10.550 37 5.2 1555 1.6 2.9 51
Are the option prices consistent with Black-Scholes?
Explain carefully why Black’s approach to evaluating an American call option on a dividend-paying stock may give an approximate answer even when only one dividend is anticipated. Doesthe answer given by Black’s approach understate or overstate the true option value? Explainyour answer.
Consider an American call option on a stock. The stock price is $50, the time to maturity is15 months, the risk-free rate of interest is 8% per annum, the exercise price is $55, and thevolatility is 25%. Dividends of $1.50 are expected after 4 months and 10 months. Show that itcan never be optimal to exercise the option on either of the two dividend dates. Calculate theprice of the option. ,
Show that the probability that a European call option will be exercised in a risk-neutral world is
N(d), using the notation introduced in this chapter. What is an expression for the value of aderivative that pays off $100 if the price of a stock at time T is greater than K?The Black-Scholes Model 261
12,23.
12.24.
12.25.
12.26.
12.27.
12,28.
12,29,
Show that S~2/"" could be the price of a traded security.Pp
ASSIGNMENT QUESTIONS
 
A stock price is currently $50. Assume that the expected return from the stock is 18% and itsvolatility is 30%. What is the probability distribution for the stock price in two years? Calculatethe mean and standard deviation of the distribution. Determine 95% confidence intervals.
Suppose that observations on a stock price (in dollars) at the end of each of 15 consecutive weeksare as follows:
30.2, 32.0, 31.1, 30.1, 30.2, 30.3, 30.6, 33.0, 32.9, 33.0, 33.5, 33.5, 33.7, 33.5, 33.2
Estimate the stock price volatility. What is the standard error of your estimate?A financial institution plans to offer a security that pays off a dollar amount equal to S} attime T.a. Use risk-neutral valuation to calculate the price of the security at time t in terms of thestock price, S, at time t. (Hint: The expected value of S? can be calculated from the meanand variance of Sy given in Section 12.1.)b. Confirm that your price satisfies the differential equation (12.15).
Consider an option on a non-dividend-paying stock when the stock price is $30, the exerciseprice is $29, the risk-free interest rate is 5%, the volatility is 25% per annum, and the time tomaturity is four months.
a. What is the price of the option if it is a European call?
b. What is the price of the option if it is an American call?
c. What is the price of the option if it is a European put?
d. Verify that put-call parity holds.
Assume that the stock in Problem 12.27 is due to go ex-dividend in 15 months. The expecteddividend is 50 cents.a. What is the price of the option if it is a European call?b. What is the price of the option if it is a European put?c. If the option is an American call, are there any circumstances under which it will beexercised early?
Consider an American call option when the stock price is $18, the exercise price is $20, the timeto maturity is six months, the volatility is 30% per annum, and the risk-free interest rate is10% per annum. Two equal dividends are expected during the life of the option with ex-dividenddates at the end of two months and five months. Assume the dividends are 40 cents. Use Black’sapproximation and the DerivaGem software to value the option. How high can the dividends bewithout the American option being worth more than the corresponding European option?262 CHAPTER 12
APPENDIX 12A
Proof of the Black-Scholes—Merton Formula
We will prove the Black~Scholes result by first proving another key result that will also be useful in
future chapters,
Key ResultIf V is lognormally distributed and the standard deviation of in V is s, then
Efmax(V — K, 0)] = E(V)N(d\) — KN(d2) (12A.1)where_ InfE(V)/K\ + s°/2SsIn[E(V)/K] — 87/2dy =
Ss
d
and E denotes the expected value.
Proof of Key ResultDefine g(V) as the probability density function of V. It follows that
E[max(V — K), 0] = [iw — K)g(V)dV (12A.2)K
The variable In V is normally distributed with standard deviation s. From the properties of thelognormal distribution the mean of In V is m, where
m = In[E(V)] — s°/2 (124.3)Define a new variableInv—Q= _—— (124.4)This variable is normally distributed with a mean of zero and a standard deviation of 1.0. Denotethe density function for Q by h(Q), so that
1n(Q) = Te
Using equation (12A.4) to convert the expression on the right-hand side of equation (12A.2) froman integral over V to an integral over Q, we get
E{max(V — K, 0)| = {. ce" — K)h(Q) dO(In K-m)/sorE{max(V — K, 0)] = | e2*™4(Q)dQ — Kf h(Q)dQ (124.5)(in K-m)/s (In K-m)/sThe Black-Scholes Model 263
Now
e210) = i & Q?+20s-+2m)/2
Vin
= LL HO- 9? 42m 7172
al
ees 2~ Ix
= ot 2a _ s)
 
2.gta-F'12
This means that equation (12A.5) becomes
OO OO
nO — s)dQ —- K| h(Q)dO (124.6)
E{max(V — K, 0)] = een(In K-m)/s
(In K—m)/s
If we define N(x) as the probability that a variable with a mean of zero and a standard deviationof 1.0 is less than x, the first integral in equation (12A.6) is
1 — N[(In K — m)/s ~ 5]* N{(—In K + m)/s + 5]
Substituting for m from equatian (17.4.3) gives
2N (sewn + 222) = Md)
Similarly the second integral in equation (12A.6) is N(d2). Equation (124.6) therefore becomesElmax(V — K, 0)] = e"**?.N(d,) — KN(d3)Substituting for m from equation (12A.3) gives the key result.
The Black-Scholes Result
We now consider a call option on a non-dividend-paying stock maturing at time 7. The strike priceis K, the risk-free rate is r, the current stock price is Sy, and the volatility is o. As shown inequation (12.22), the call price, c, is given by
c=e 7 Elmax(Sp — K, 0)] (124.7)
where Sp is the stock price at time J and E denotes the expectation in a risk-neutral world. Underthe stochastic process assumed by Black