-Scholes, S; is lognormal. Also from equations (12.3)and (12.4), E(Sr) = Sye’ and the standard deviation of In Sy is oT.
From the key result just proved, equation (12A.7) implies that
c= eT [Sye"” Md) — KN)orc= SpN(d,) — Ke N(dy)264 CHAPTER 12
 
where .
_ InfE(S7)/K] + 0°T/2=P_ In(So/K) + (r+? /2)T= aR
qd
andg, = MEG)/K1- @°T/2= GF_ In(So/K) + (x - 0° /2)T= ORThe Black-Scholes Model 265
APPENDIX 12B
Exact Procedure for Calculating Values of American Calls onDividend-Paying Stocks
The Roll, Geske, and Whaley formula
e RO, Greske, and Wha!
single dividend D, at time t is
for the value of an American call o
C= (Sp — Dye )N(by) + (Sy — Dey (ay, bi; -\t)
- Ke" Mas, —by; — i) ~(K — Dye" (bs) (12B.1)
wherea, —lnllSo = Die VK + (4 F/T= oVTay =a; — oVTp, — nlSo = Die 8") + +.07/2)4~ oJby = by — oth
The variable o is the volatility of the stock price less the present value of the dividend. The functionM(a, b; p), is the cumulative probability in a standardized bivariate normal distribution that thefirst variable is less than a and the second variable is less than b, when the coefficient of correlationbetween the variables is o. We give a procedure for calculating the M function in Appendix 12C.The variable S* is the solution to
c(S*) = S*4+D,-K
where c(S*) is the Black-Scholes option price given by equation (12.20) when the stock price is S*and the time to maturity is 7 —7¢,. When early exercise is never optimal, S* = oo. In this case,b, =b, = —oo and equation (12B.1) reduces to the Black-Scholes equation with Sg replaced bySy- Dye". In other situations, S* < oo and the option should be exercised at time t; whenS(t) > S*+ Di.
When several dividends are anticipated, early exercise is normally optimal only on the final ex-dividend date (see Section 12.13). It follows that the Roll, Geske, and Whaley formula can be used -with S) reduced by the present value of ail dividends except the final one. The variable D, shouldbe set equal to the final dividend and t, should be set equal to the final ex-dividend date.266 CHAPTER 12
APPENDIX 12C
Calculation of Cumulative Probability in Bivariate NormalDistribution
As in Appendix 12B, we define M(a, b; e) as the cumulative probability in a standardized bivariatenormal distribution that the first variable is less than a and the second variable is less than b, whenthe coefficient of correlation between the variables is . Drezner provides a way of calculating. M(a, b; p) to an accuracy of four decimai places.’® Ifa <6, 6b <0, and p< G, then
V1 — 4ca
M(a, b; p) = A;A;f(B;, B;)ij=lwhereF(x, y) = exp[a(2x — a’) + b'Qy — bY + 2p(x — ay — BY]i a f b
—.-—, Pp = ——_V20 — p) V20 = p°)A; = 0.3253030, Ap =0.4211071, A; = 0.1334425, Aq = 0.0060374323
B, = 0.1337764, By = 0.6243247, B, = 1.3425378, By = 2.2626645
In other circumstances where the product of a, b, and p is negative or zero, one of the followingidentities can be used:M(a, b; p) = N(a) ~ M(a, —b; —p)
M(a, b; p) = N(b) — M(—a, b; —p)M(a, b; p) = N(a) + N(b) — 1 + M(~a, ~b: p)In circumstances where the product of a, b, and 9 is positive, the identityM(a, b; p) = M(a, 9; 2) + M(B, 0; p2) — 8
can he used in conjunction with the previous results, where
 
 
 
(pa = b)senta) (ob—a)sen(d) 1 ~ sana) sent)Va —2pab +B? Va? —2pab +b?" 4withsan(t) = [" ifx>0e -l ifx<0
'° Z. Drezner, “Computation of the Bivariate Normal Integral,” Mathematics of Computation, 32 (January 1978),277 79. Note that the presentation here corrects a typo in Drezner’s paper.Omura Ayr rem «a4
CHAPTER 13
OPTIONS ON STOCK INDICES,CURRENCIES, AND FUTURES |
 
In this chapter we tackle the problem of valuing options on stock indices, currencies, and futurescontracts. As a first step, we extend the results in Chapters 8, 10, and 12 to cover Europeanoptions on a stock paying a known dividend yield. We then argue that stock indices, currencies,and futures prices are analogous to stocks paying a known dividend yicld. This enables theresults for options on a stock paying a dividend yield to be applied to value options on theseother assets. ,
13.1 RESULTS FOR A STOCK PAYING A KNOWNDIVIDEND YIELD
—
 
Consider the difference between a stock that pays a dividend yield at a rate g per annum and anotherwise identical stock that pays no dividends. Both stocks should provide the same overallreturn (dividends plus capital gain). The payment of a dividend causes the stock price to drop bythe amount of the dividend. The payment of a dividend yield at rate g therefore causes the growthrate in the stock price to be less than it would otherwise be by an amount g. If, with a dividendyield of g, the stock price grows from Sy at time zero to Sp at time 7, then in the absence ofdividends it would grow from Sy at time zero to Sye” at time T. Alternatively, in the absence ofdividends it would grow from Spe‘ at time zero to Sp at time T.
This argument shows that we get the same probability distribution for the stock price at time Tin each of the following two cases:
I. The stock starts at price Sy and pays a dividend yield at rate gq.2, The stock starts at price Sye~®” and pays no dividend yield.
This leads to a simple rule. When valuing a European option lasting for time T on a stock paying aknown dividend yield at rate g, we reduce the current stock price from Sp to Spe *” and then valuethe option as though the stock pays no dividends.
Lower Bounds for Option Prices
Asa first application of this rule, consider the problem of determining bounds for the price of aEuropean option on a stock providing a dividend yield equal to g. Substituting Sye~*” for So in
267268 CHAPTER 13
equation (8.1), we see that the lower bound for the European call option price, c, isc > max(Sye"” — Ke’, 0) (13.1)
We can also prove this directly (see Problem 13.40) by considering the following two portfolios:
Portfolio A: one European call option plus an amount of cash equal to Ke7?
Portfolio B: e~ shares, with dividends being reinvested in additional shares
To obtain a lower bound for a European put option, we can similarly replace Sy by Sye7? iequation (8.2) to getp > max(Ke? — 50 0) (13.2)
This result can also be proved directly (see Problem 13.40) by considering the following twoportfolios:
Portfolio C: one European put option plus e~” shares, with dividends on the shares beingreinvested in additional shares
Portfolio D: an amount of cash equal to Ke~”
Put-Call Parity
Replacing Sp by Sge~” in equation (8.3), we obtain put-call parity for a stock providing a dividendyield equal to g:ob Bel op 1 6a (132.2c+Ke = p+ Soe (13.3)This result can also be proved directly (see Problem 13.40) by considering the following twoportfolios:
Portfolio A: one European call option plus an amount of cash equal ta Ke” rT
Portfolio C: one European put option plus e~” shares, with dividends on the shares being
reinvested in additional shares
13.2. OPTION PRICING FORMULAS
 
By replacing S) by Soe” in the Black-Scholes formulas, equations (12.20) and (12.21), we obtainthe price c of a European call and the price p of a European put on a stock providing a dividendyield at rate g as
c= Sye*" N(d) — Ke-™ N(dp) (13.4)p = Ke N(—dy) — Sge~4" N(—d)) (13.5)
—gTnS ) =In n%_ gr
Since
 Options on Stock Indices, Currencies, and Futures 269
it follows that d, and d> are given by
_ In(So/K) + r= 9 +.0°/2T
a oJ/T2a, = OU #4 [MT _ 4, oF
These results were first derived by Merton.’ As discussed in Section 12.13, the word dividendshould be defined as the reduction of the stock price on the ex-dividend date arising from anydividends declared. If the dividend yield is not constant during the life of the option, equa-tions (13.4) and (13.5) are still true, with gq equal to the average annualized dividend yield duringthe life of the option.
Risk-Neutral Valuation
Appendix 13A derives, in a similar way to Section 12.6, the differential equation that must besatisfied by any derivative whose price, f, depends on a stock providing a dividend yield g. Like theBlack-Scholes differential equation (12.15), it does not involve any variable affected by riskpreferences. Therefore, the risk-neutral valuation procedure, described in Section 12.7, can beused. In a risk-neutral world, the total return from the stock must be r. The dividends provide areturn of g. The expected growth rate in the stock price, therefore, must be r — g. The risk-neutralprocess for the stock price is
' dS = (r — q)Sdt +0S dz (13.6)
To value a derivative dependent on a stock that provides a dividend yield equal to g, we set theexpected growth rate of the stock equal to r— gq and discount the expected payoff at rate r. Toapply this approach to valuing a European call option, note that the expected stock price at time Tis Spe". Equation (12A.1) gives the expected payoff from a call option as
ef -9T § Nd) — KN(d)
where d, and d are defined as above. Discounting at rate r for time T leads to equation (13.4).
Binomial Trees
We now examine the effect of a dividend yield equal to g on the binomial model in Chapter 10.Consider the situation in Figure 13.1, in which a stock price starts at Sy and, during time 7, moveseither up to Sow or down to Spd. As in Chapter 10, we define p as the probability of an upmovement in a risk-neutral world. The total return provided by the stock in a risk-neutral worldmust be the risk-free interest rate, r. The dividends provide a return equal to g. The return in theform of capital gains must be r — q. ‘This means that p must satisfy
pSou + (1 — p)Syd = Sye?"
' See R. Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science,4 (Spring 1973), 141-83.270 CHAPTER 13
Sout
an
Sodfa
Figure 13.1 Stock price and option price in one-step binomial treewhen stock pays a dividend at rate g
oreral _g
p= u-—d
(13.7)This is the same equation as in Chapter 10 with r replaced by r — g. As in Chapter 10, the value ofthe derivative is the expected payoff in a risk-neutral world discounted at the risk-free rate, so that
fae" pf, + (1p) fil (13.8)
Example 13.1 Suppose that the initial stock price is $30 and the stock price will move either upto $36 or down to $24 during a six-month period. The six-month risk-free interest rate is 5% and thestock is expected to provide a dividend yield of 3% during the six-month period. In this case uv = 1.2,d = 0.8, and (0.05-0.03)x0.5 _ 9 g
p=" a = 0.5251Consider a six-month put option on the stock with a strike price of $28. If the stock price moves up,the payoff is zero; if it moves down, the payoff is $4. The value of the option is therefore
—O08¥05r, cane 4 ma a sone ~ JU.dZ5L X U+U.4/49 X 4] = 1.83
13.3 OPTIONS ON STOCK INDICES
Options on stock indices trade in both the over-the-counter and exchange-traded markets. Asdiscussed in Chapter 7, some of the indices track the movement of the stock market as a whole.Others are based on the performance of a particular sector (e.g., mining, technology, or utilities).
Quotes
Table 13.1 shows quotes for options on the Dow Jones Industrial Average (DIX), Nasdaq (NDX),Russell 2000 (RUT), S&P 100 (OEX), and S&P 500 (SPX) index options as they appeared in theMoney and Investing section of the Wall Street Journal on Friday, March 16, 2001. All theoptions trade on the Chicago Board Options Exchange and all are European, except the contracton the S&P 100, which is American. The quotes refer to the price at which the last trade was madeon Thursday, March 15, 2001. The closing prices of the DIX, NDX, RUT, OEX, and SPX onMarch 15, 2001, were 100.31, 1,697.92, 452.16, 600.71, and 1,173.56, respectively.Options on Stock Indices, Currencies, and Futures 271
 
Table 13.1 Quotes for stock index options from The Wall Street Journal, March 16, 2001
 
        
 
  
May 2500 ¢ 2Thursday, March 15, 2001 Apr 1200p NASDAO-100 = 10 9g0 | May 2500p aVolume, last, net change and open interest | May 1200p v7 8 - 1 a1 Mar see 19for all contracts. Volume figures are unofficial.| Apr 1300p 12 - 6 444 | Mar 00. 599Open interest reffects previous trading day. p-Put ha {400p 622910 = mo 1,827 | AD ame ;call y 1400p = 5B 128 ie p 8Ap = 1500p = 104-53 + 517 | Ma 2700p 332NET OPEN | Mar 1550p 0 «(0 = 175 2 in 2800 ¢ 2726STRIKE VOL. LAST CHG, INT. | Apr = 1550p 6 42 = 160 208 | May $9 2c 26J er tpt «ms Gag] Sun 2800 1.087CHICAGO we Be, LI ee ffot ele Sa ssr ¢ - sess "01 NOUS AE Ma 160p 38BOS Ot 8d 0dun fp |? ~ 008 tS ie 160 p 2% - 597dun Mp = 10, HB - OHS. 909 7c 27 85 263 | Ma 4506Ay pS we | Mar 1700p 1,058 25 .. 2491 Ma 450wo Mp t= 0 1,096 1700p = 10 133-180 BA p58jo Bok TT Ney DH - 1 adaeStmo —a wp ad bebe oo s | dun i7c 20 -i7 20 | Mar 460 a 19 4Jun Op 12 190 $080 un 1700p 3 180 799 | gx . 1H ‘ee -Sep 0p 3 2-3 Te 750c «| 058k 0 $60p 201 2280 +Apr Sp 1787 (OS - Of 564) Ma 1750p = 1t 65+ 280M in 470¢ 10 050dun Sep 51 — OS 2.881 | ap 17500 6 HD = 15 6 4710p 5002480Sep 2p 58 3 + OS 503 ar 7p 363 152 + 30 500p 100 8+Mar Be 8 4 + OO Id 1e00c 480 24Bsi‘ UO 2K 500 ¢ 10 70Mar Hp 170 HS — OS 12199 | Ma = 1800p «= 100+ 15 510 3 3Apr Ke 6 580 - 80 3 | Ap 1800¢ 1 1200 +2 Open int,Apr %p S72 5 — OD 5,670 1900p 80 170 + 15 Open Int.Jun %e 3 7H ~ 10 48ST un 1800. 315 4 15hn Bp 6B HO — HO AGE? | ten 1800p S628 $ & P 106(0EX)Sp 6p 2B 4 » 6374] Ma 1660¢ 536.125 78 sp 32 4515Me = He MM - OB Mar 1850p «215880 + 9380 500p 24 ‘88ar Sip i os 1502 | Ay 1850 ¢ 2 «7380 — 3920 500 1 550 3Ay = Bc HAH ODT ie 1850p 6 19 - 1 0p 1425 — O88 1.233by Kp 836220 ~ 6469 | Ma 900c 369 089 - 3 540p a7? — OS 3,610fr «= 0c) 2.323070 = 08389 | 1800p) 190 +60 5340p 28 5 - 1 408Mar 1090p 4550 0% - O70 14.425 1 Ae = 1900 «= 202 6g30 — 1880 540 p 10 11 — 240 538Ay = 100¢ 483380 + DAG 1900p «Bat + 16 550 ¢ 5 $2 + 610 7Ay 100p 1,502 85 — 05 10,208 | May 1900 1 142 + 3380 5p 749 OF - 08 9503dun 1c 112 6 ~ 1292 | dun 19006 % 10 -% 550p 7 70 - 10 BOBdun 1p 3119 410 - 0% 7207 | Mar «t= 113sQtO at 555 p 2 720 oe “0Sep 100¢ 2 7 60 30} Ma 1950p 82 20 +2 S60p = 857 I-00 8,327Sep 1p CG we 4,085 1950¢ 5 8 + 5 560 ¢ 1 5950 aMar 102c 47810 O15 381 | Ane «= 9950p BMH OS 560p 12420 110286{2p 16 1% — ow Hee | iy mek 1) es 580p 122 1$10 + 734Apr 2c 148 2 OSG Mar «= Op) 103 + 5 S70p 2136 0% =Apr Wp 103 3-0 OAT Tar 208 28-1880 $70¢ a7 +dun {2c 40 - 040 2,295 | Ane = 2050p = 107, 308 — 3480 S70p 189 1070 -wun 2p 513+ OM 2313 Tay 2050 { 86 - 2 575 4 1150Ma 103 9 OS SATO | Mar) 2t00p 3 395 + 45 S80p 216 1 ~Mar 1038p = 305. 8D DST (Ay =O 5600p 320 teMa 1104p 740 = 08,693 2100p = 100 385 + 15 S80p 4170104c 7 180 + 010 1696 | Jun = 2100 1 7g60 ~ 3440 560c¢ ©9122, 4850 ~A (Op 19710 080 13,625 | Ma «= 2150 1 0 - Ow 0p 13 AKdun 104 3 + 080 2.484 | Mar 2150p 8 370 «+ 70 585 p 1 1430hin 104p = 280+ Geo 14g i 2150p 4 455 + 408 be 1ST if =Sop 10 p 1 7 + 040 996 | Mar 22006 6 Os — 085 5p 3264 1M -1055p 127, 4-0 1,536 | Mar 2200p 12 435 ~ 6000 590¢ 2% AM +Mar 106 p Bi 6 + «05792 | Apr = 200 3 1550 — 490 590 p & & -Ay = 10606371 ROL Ae «= Damp 1470 + 185 590 p 2 2Ay 6p 4410 OW 43,208 aay 2200p 300 435-47 595p 27 1920dun 106 ¢ 8 280 + 020243 | Jun B00 1 6 +5 800c «6,508 | 5mun 1106p = 61730 + 080698 | un 2000p = 52 aH 2H 600p 8644 40Mar 107 p 2 68 - 010852 | Mar 20800 15S OS 800c = 303230 +Mx = 1088p 2d 8 3982 | Mar 2250p 7 4 + 6 B00p = 8B NYAp 105158 + HD 1.226 | Ar He «=| 710 600p 105 4KAp 0p BO 70,597 | ar «= 25006 3 10 4 4 600 ¢ 1 SO +dun 1068p = 53 8 1917 | Mar 2300 p 1 525 + 27 0p 19 BHSop 1108p tt 80 + 0 HB a 30D BT S00 605 4900-285Mar 110p 10780 =~ O10 BOD | Apr = 2300 p 4 58310 + 188 605p 1901 70 -{10¢ 1,022 05 7792 | Jun 2300p 1 $5590 + 080 650 | 261+for = 110p 58930 ~ O376 Mar «2350 «= St BEKO + 220 605p 23-2380Jun 110p 4 - 2400 p 5 685 + 10 610c 6920 175 ~Ma = 112p tt - 2400 ¢ 2 jo 4 20 610p 2738 1 -Aor tt2c 00 + moc 86 BD 610c 9B 7t01 - 2400 p 4 61590 — 6910 6p 8691 - 2450 p 2 65 - 5 610 4 31509.262 2500 p 3 60 + 5 5c 1773 OH87 oc 8 12H 615 6822080 +
 
    
 
 
 
(continued on next page)272 CHAPTER 13
 
Table 13.1 (continued)
 
        
  
 
  
 
    
Ay = 50+ OH OD 8 a p sgisen My 1275p 9 108 + 1 38HAo 6i5p 2 — - 5 315] Jun Tc 300 - 19 = OB] un 12750 119 + 0 6297is «6c 4149 3pat | Jun 75Dp M4 — OB 555| Jun Ho75p aa + 58 885eee ere ee ae RIB ee a lyc + n B = lar +Pe ee eae Be als able BP Beck oeMy p18 ag oo | Ay $50p «Bi OOS Teh ay — ja00p B75! 1238 — 1200 GUDhn He 06 OH 985 | dun $500 70 + 1m 68021 May 900 165 est > om 4in fp 8 fo Go tell ay | Sep 125 oe to ee Op “74 124 - 4 536Ma 625c BIO — tye | Nr Be 8 HL NO Hae un ODE 438s 8) 7770Ma 625p 55 «5 BO Mar) Hp 10] — OS 5294 | un 1300p HHS Sm 1356)Aw Bo] + O80] dun HEN CHS 94) Mar 0p 3 15,8625 p 7 30M + 430 182 | Jun $5 p 3 11 - = 4940] Mar 1325 ¢ 23010 + 08547198ie Ge tg Os TO ae 5p M4 sid 74 | Mar 3D Tw 10 aDMa 6830p 98 = D105 ie 1025p = 200 US - OB | Apr «| 1199S - OM 14035Apr 0c 8156 HH OMB he 1025p 22 G6 = 1904829 | Apr 1325p = Ot «14680 — 1880 2854top ea ota | Mar 105d «32D — Os Seu lthy ae oFee ale Be ad ie tl ee SET aeMe Sse 1a Os > Oe opar| kn BOD 146 To + GYD T hn Hsp 10H 1B 10 S688Ma  635p = 456-3520 ~ 280 5 he imep 3 it, ~ on pa Nar 0p va ir — 50 B28i rat Dp 7 i aa ep 3 a + or Pa Ap 1100¢ 1 9260 + 160 S? ha Sp 6113 173 + 3H ‘iPee SOP Se ee Boe le Bk el- - ‘ c 7 >ae ee ee a ee0 950 n -