i - :ey fae io - 4 a Mar 1125p 1903020 = 14.529 e tgp 5) 1% = 40 B14May 0p me hohe orn ao samt eet le ORRwn 8p Smt et dan ipe ‘ooo st + ge mi ws rap st 2 =H 11152410 5 lar ~ fWe eee Ge el wer avy 4a I) — oo WIA) eo ie BeeAp 4S 4 5 = 4 448 | Apr tte 5 + 3 489 | Jun 1400p 79 20680 - 1550iy lbp ts tes] Ae Nb Aer Tw 0 aR esp DHMa — 650c 1215 = OSH t51 | May 1150 69 +5 105 Ar 4c 2 0 + OS 9371Mx 6850p 129, B00 gto. 4o7 | May = 1150p 7741 6 ~ 3 CBB] May 145 2 21 + 83Ap = 650 «OD 420 = 00 tg] dun 1150 | 604 | un 1425p 125 25+ 4 68HOp 32 = Zoe] un 1180p 72 42 — 5 938 Me 1p | 8k 71Stay 650c 7 4 i Mar 1175¢ 2260 280 — 64 = 2960] Apr 1450p 3 269 + «(180wp te Me tp 74l 6 — wo Baal ay fae | ow 2 Oe50. Apr 1175 6 tt 670 + 0h 46 1450 4 65 we aie ee Oe hy ip 2 Ys 4 72 Mey itp, t - ;ime an ———————te wpe sp - 6° “a | o tep aie ie - di ssi] LEAPS-LONG TERApr a an z O50 2959 | 44755 «Year san — sn 7'oR0vy OC 2a) go 2m | Mar AME BHF 2 BBL Ay agg MM oglar p - fyar ee ge ET aw 2006 2g) Beko — om Saar | Bee - @foc 18) 0 - Os Ah | A 1200p 4, 4% - 4 © 16070 | Dec 02 -4bx Be age | May 1200 4 + 4 1565 | Doc 02 - &Ma -670p «GAL SHG | May 1200p ALB - 44570 | Dec 02 + OseBe GS Seo be lie a ee aL elasjun p 6 = ,fe fee el Mar ete | SSt 0 — PMe ee att “tes gag | Max 2p B+ 801483 | ‘Ma 7p HS Tg - GH Ta] Mat WDC OSI al Vat S  GpenAe cB] HS OS ag | Ba 962 | Put Vel. 4 Openz y 1205 OS — 0% 12754Re et bo wh f Mar 1p th 8 8L6r 500 -¢BoB ble lh Be Bie 26, Wee oh 2 Tean p : 1950 + 510 ‘fap | Pee p +Oe Ge ale Be be Bee ih, 3 ao aehe eae (tes = os gat | un 1B 10 - 4 ROBB occ om HHO pot am + fd 738sHp 108 = 10 353] tun 2p 8 8 12580) dec O3 110 p 10 ge — Ow BeeWy Oe atom 2 Go ge] Ma De DO OF 10889) tor ty bp 5D a tw RAIDay c z Mar 1290p 5650 — 1750 3.435 70May 690p 10 9 + 8 47 Gee 02 12 5 8 {1 + Gao AG68wa 700 qn. 40 War iaibe Hope | OO SBT) Dee OF ap | § tm — om Sah7ce a ee re ee erto = 1066 Qo + 00s Ay fae 545 10 — 1 780 | Oe Ie , # feo > go Baietee tO oe ols 10ag} Ar | Dp «SB Bw - Bm 9EB3 I Oe or 135 bw tt BIAor Oc BL GMI Mey ioe 10 mo + im aloe gt ap > ‘Be 5 be fatfe le Be Take |S] May 0p 6 GH BBY Dee Oy $4) 155 200 + O10 4Z5D5we tts os maple Be BY ERI ne we POY ae nimc - dun 1250p - k Dec 01 145 2 460 5to Rp K ‘an +4, zl Mar 1p % on + bi oe bec Ot 145 p @ ne t 50 3.430- lar c - R 2g70 — 743wy pS Cet + a0 Mar «osc tgs om pelos eb Sf aeMay 7600 4 = 5H Mar 1275p «HP 103. — 4” (22621 I Dec OL $80 ¢ 3 OS 56Mi 3 4 12750 | 4481 6 + OW 14960) Dec 02 180 p 10 SD +Wee ho 555 Gat BAe @ Oper¢ + ‘ pen
 
Source: Reprinted by permission of Dow Jones, Inc., via copyright Clearance Center, Inc.© 2001 Dow Jones & Company, Inc. All Rights Reserved Woridwide.Options on Stock Indices, Currencies, and Futures 273
One index option contact is on 100 times the index. (Note that the Dow Jones index used forindex options is 0.01 times the usually quoted Dow Jones index.) Index options are settled in cash.This means that, on exercise of the option, the holder of a call option receives S$ — K in cash andthe writer of the option pays this amount in cash, where S is the value of the index at the close oftrading on the day of the exercise and K is the strike price. Similarly, the holder of a put optionreceives K — S in cash and the writer of the option pays this amount in cash.
Example13.2 Consider the April put option contract on the S&P 100 with a strike price of 620 inTable 13.1. This is an American-style option and expires on April 21, 2001. The cost of one contractis indicated as 29.10 x 100 = $2,910. The value of the index at the close of trading on March 15,2001, is 600.71, so that the option is in the money. If the option contract were exercised, the holderwould receive (620 — 600.71) x 100 = $1,929 in cash. This is less than the value of the contract—indicating that it is not optimal to exercise the contract on March 15, 2001.
Table 13.1 shows that, in addition to relatively short-dated options, the exchanges trade longer-maturity contracts known as LEAPS, which were mentioned in Chapter 7. The acronym LEAPSstands for Long-term Equity AnticiPation Securities and was originated by the CBOE. LEAPS areexchange-traded options that last up to three years. The index is divided by five for the purposes of
soting the strike nrice and the ontion price me contract is an option on imes one. Soquoting the strike price and the option price. One contract is an option on 100 times one-fifth of
the index (or 20 times the index). LEAPS on indices have expiration dates in December. Asmentioned in Chapter 7, the CBOE and _ several other exchanges also trade LEAPS on manyindividual stocks. These have expirations in January.
The CBOE also trades flex options on indices. As mentioned in Chapter 7, these are optionswhere the trader can choose the expiration date, the strike price, and whether the option isAmerican or European.
Portfolio Insurance
Portfolio managers can use index options to limit their downside risk. Suppose that the value of anindex today is Sy. Consider a manager in charge of a well-diversified portfolio whose beta is 1.0. Abeta of 1.0 implies that the returns from the portfolio mirror those from the index. If the dividendyield from the portfolio is the same as the dividend yield from the index, the percentage changes inthe value of the portfolio can be expected to be approximately the same as the percentage changesin the value of the index. Each contract on the S&P 500 is on 100 times the index. It follows thatthe value of the portfolio is protected against the possibility of the index falling below K if, foreach 1005p dollars in the portfolio, the manager buys one put option contract with strike price K.Suppose that the manager’s portfolio is worth $500,000 and the value of the index is 1,000. Theportfolio is worth 500 times the index. The manager can obtain insurance against the value of theportfolio dropping below $450,000 in the next three months by buying five put option contractswith a strike price of 900. To illustrate how this works, consider the situation where the index dropsto 880 in three months. The portfolio will be worth about $440,000. The payoff from the optionswill be 5 x (900 — 880) x 100 = $10,000, bringing the total value of the portfolio up to the insuredvalue of $450,000.
When the Portfolio’s Beta Is Not 1.0
If the portfolio’s returns are not expected to equal those of an index, the capital asset pricing modelcan be used. This model asserts that the expected excess return of a portfolio over the risk-free274 CHAPTER 13
Table 13.2. Relationship between value ofindex and value of portfolio for beta = 2.0
 
Value of index Value of portfolioin three months in three months (8)1,080 570,0001,040 530,0001,000 490,000960 450,000920 410,000880 370,000
interest rate equals beta times the excess return of a market index over the risk-free interest rate.Suppose that the $500,000 portfolio just considered has a beta of 2.0 instead of 1.0. Suppose furtherthat the current risk-free interest rate is 12% per annum, and the dividend yield on both the portfolioand the index is expected to be 4% per annum. As before, we assume that the S&P 500 index iscurrently 1,000. Table 13.2 shows the expected relationship between the level of the index and thevalue of the portfolio in three months. To illustrate the sequence of calculations necessary to deriveTable 13.2, Table 13.3 shows what happens when the value of the index in three months proves tobe 1,040.
Suppose that Sy is the value of the index. It can be shown that, for each 1005, dollars in theportfolio, a total of beta put contracts should be purchased. The strike price should be the valuethat the index is expected to have when the value of the portfolio reaches the insured value.Suppose that the insured value is $450,000, as in the case of beta = 1.0. Table 13.2 shows that theappropriate strike price for the put options purchased is 960. In this case, 00S) = $100,000 andbeta = 2.0, so that two put contracts are required for each $100,000 in the portfolio. Because theportfolio is worth $500,000, a total of 10 contracts should be purchased.
Table 13.3 Calculations for Table 13.2 when the value of the index is1,040 in three months
 
Value of index in three months 1,040Return from change in index 40/1,000, or 4% per three monthsDividends from index 0.25 x 4 = 1% per three monthsTotal return from index 4+ 1=5% per three monthsRisk-free interest rate 0.25 x 12 = 3% per three monthsExcess return from index
over risk-free interest rate 5—3=2% per three monthsExcess return from portfolio
over risk-free interest rate 2x 2=4% per three monthsReturn from portfolio 3+4=7% per three monthsDividends from portfolio 0.25 x 4= 1% per three monthsIncrease in value of portfolio 7—1=6% per three months
Value of portfolio $500,000 x 1.06 = $530,000Options on Stock Indices, Currencies, and Futures 275
To illustrate that the required result is obtained, consider what happens if the value of theindex falls to 880. As shown in Table 13.2, the value of the portfolio is then about $370,000. Theput options pay off (960 — 880) x 10 x 100 = $80,000, and this is exactly what is necessary tomove the total value of the portfolio manager’s position up from $370,000 to the required levelof $450,000.
Valuation
Tn valuing index futures in Chapter 3, we assumed that the index could be treated as a security payinga known dividend yield. In valuing index options, we make similar assumptions. This means thatequations (13.1) and (13.2) provide lower bounds for European index options; equation (13.3) is theput-call parity result for European index options; and equations (13.4) and (13.5) can be used tovalue European options on an index. In all cases Sp is equal to the value of the index, o is equal to thevolatility of the index, and q is equal to the average annualized dividend yield on the index during thelife of the option. The calculation of g should include only dividends whose ex-dividend date occursduring the life of the option.
In the United States cede dates tend to occur during the first week of February, May.August, and November. At any given time, the correct value of g is therefore likely to depend onthe life of the option. This is even more true for some foreign indices. For example, in Japan allcompanies tend to use the same ex-dividend dates.
Exampiet3.3 Consider a Luropean call option on the S&P 500 that is two months from maturity.The current value of the index is 930, the exercise price is 900, the risk-free interest rate is 8% perannum, and the volatility of the index is 20% per annum. Dividend yields of 0.2% and 0.3% areexpected in the first month and the second month, respectively. In this case Sy = 930, K = 900,r=0.08, o=0.2, and T7=2/12. The total dividend yield during the option’s life is0.2+0.3 = 0.5%. This is 3% per annum. Hence, g = 0.03 and
In(930/900) + (0.08 — 0.03 + 0.22/2) x 2/12
 
 
 
d 0.5444| 0.22/12_ _ 2m(930/900) + (0.08 ~ 0.03 - 0.2/2) x 2/12 _ 9 gang
 
 
0.242/12N(d,) = 0.7069,  N(da) = 0.6782so that the call price, c, is given by equation (13.4) as© = 930 x 0.7069e° 87/1? _ 900 x 0.678270 87/1? — 51,83
One contract would cost $5,183.
If the absolute amount of the dividend that will be paid on the stocks underlying the index (ratherthan the dividend yield) is assumed ‘to be known, the basic Black-Scholes formula can be usedwith theapproach recommended i in Chapter 12 fora stock paying known dividends. However, the approachmay be difficult to implement for a broadly based stock index because it requires a knowledge ofthe dividends expected on every stock underlying the index.
In some circumstances it is optimal to exercise American put options on an index prior to theexercise date. To a lesser extent, this is also true of American call options on an index. American
1 stock price being reduced by the present value of the divi
i stock price being reduced the present value oF the
    276 CHAPTER 13
stock index option prices are therefore always slightly more than the corresponding Europeanstock index option prices. We will look at numerical procedures for valuing American indexoptions in Chapter 18.
13.4 CURRENCY OPTIONS
 
 
European and American options on foreign currencies are actively traded in both the over-the-counter and exchange-traded markets. The Philadelphia Stock Exchange has been tradingcurrency options since 1982. The currencies traded include the Australian dollar, British pound,Canadian dollar, Japanese yen, and Swiss franc. For most of the currencies the PhiladelphiaExchange trades both European and American options.
For a corporation wishing to hedge a foreign exchange exposure, foreign currency options are aninteresting alternative to forward contracts. A company due to receive sterling at a known time inthe future can hedge its risk by buying put options on sterling that mature at that time. Thestrategy guarantees that the value of the sterling will not be less than the exercise price whileallowing the company to benefit from any favorable exchange-rate movements. Similarly, acompany due to pay sterling at a known time in the future can hedge by buying calls on sterlingthat mature at that time. The approach guarantees that the cost of the sterling will not exceed acertain amount while allowing the company to benefit from favorable exchange-rate movements.Whereas a forward contract locks in the exchange rate for a future transaction, an option providesa type of insurance. This insurance is, of course, not free. It costs nothing to enter into a forwardtransaction, whereas options require a premium to be paid up front.
Quotes
Table 13.4 shows the closing prices of some of the currency options traded on the PhiladelphiaStock Exchange on Thursday, March 15, 2001, as reported in the Wall Street Journal of Friday,March 16, 2001. The precise expiration date of a foreign currency option is the Saturday precedingthe third Wednesday of the maturity month. The sizes of contracts are indicated at the beginning ofeach section of the table. The option prices are for the purchase or sale of one unit of a foreigncurrency with U.S. dollars. For the Japanese yen, the prices are in hundredths of a cent. For theother currencies, they are in cents. Thus, one call option contract on théseuro with exercise price90 cents and exercise month June would give the holder the right to buy 62,500 euros for 56,250(= 0.90 x 62,500) U.S. dollars. The indicated price of the contract is 2.34 cents, so that onecontract would cost 62,500 x 0.0234 = $1,462.50. The spot exchange rate is shown as 88.15 centsper euro.
Valuation
To value currency options, we define § as the spot exchange rate (the value of one unit of theforeign currency measured in the domestic currency). We assume that S follows a geometricBrownian motion process similar to that assume for stocks in Chapter 12. In a risk-neutral worldthe process is
dS = (r —rs)Sdt + oSdz
where 7 is the domestic risk-free interest rate, rp is the foreign risk-free interest rate, and o is theexchange rate’s volatility.Options on Stock Indices, Currencies, and Futures 277
 
Table 13.4 Currency option prices on the Philadelphia Exchange
 
 
  
from the Wall Street Journal on March 16, 2001PHILADELPHIA EXCHANGE OPTIONSCALLS PUT CALLS PUT CALLS PUTVOL, LAST VOL. LAST VOL. LAST VOL. LAST VOL. LAST VOL. LASTDlr 54.26 | JYen 93.49} 96 Mar a we 5 BRB50,000 Australian Dollar EON Exropean, et ie ow Eeropean ste, Euro 68.18
St Ant 4 ~ = 355) 62500 Euro-cents per unit.
AD? 5426 | J 93.15] 5g age 0 04350,000 Australian Dollars-| “European Style. 20) 000J.Yen-100ths of a cent per ait _
52 Mar . 427i] "B19 Mar 030} 99 lun 2 29 1 238Dour 5426 | 8250 Apr 2 10 2 144] 94 Mar me S86550,000 Australian Dollars-cents fet nit 83 Apr . he 5 151) 114 Mar ws ” 2 23.56
oni dun 1 ar Net sony ' sive $92.53 | SFranc 57.83
. 200 .Yen-Evrepean: ji |S00 CoadanQolars-European Ste 7 tar HD $80, | Udy SRS Frame EON Europe she
65 Mar om 0B] 19 Sep NO 640 Egg , 183
6750 Mar 0 335) 82 ar 2 09 | 2] SFranc a1
879 Sep ge BTA 2 195 ., __... 62.500 Swiss Francs-European Style.
Cole 66.48 | Euro 06.18] 58 Apr 16 1.3050,000 Canadian Dollars-cents oe unit. 62,500 Euro-European style 60 Apr 16 0.34ete ag) e ie: g dele vslar ae . 4 un we . 4 ie 77450: Mar 0 ina) 98 un 5 748 | S2500 Swiss Francs-ents per utva 20 1086) 104 Mar 8 08 Abr ;66.48 | Euro 885] 60 Sep 2 1samo onan Dulas eats r unit 62,500 Euro-European style. Call Vol, 1.221 Open Inti) 92 Apr 2 073. ” an E Put Ve 3.926 Open Int.
 
 
Source: Reprinted by permission of Dow Jones, Inc., via copyright Clearance Center, Inc.© 2001 Dow Jones & Company, Inc. All Rights Reserved Worldwide.
This stochastic process for an exchange rate is the same as that in equation (13.6) with g = r;. Thisis because, as noted in Section 3.11, a foreign currency is analogous to a stock providing a knowndividend yield. The owner of foreign currency receives a “dividend yield’’ equal to the risk-freeinterest rate, ry, in the foreign currency. Because the stochastic process for a foreign currency is thesame as that for a stock paying a dividend yield equal to the foreign risk-free rate, the formulas inSection 13.] and 13.2 are correct with q replaced by r;. The European call price, c, and put price, p,are therefore given by
¢ = Sye!" N(d,) — Ke" N(d>) (13.9)
p= Ke" N(—dy) — Spe"! N(—d,) (13.10)where Sp is the value of the exchange rate at time zero, and_ In(Sg/K) + (r= rp +07 /2)T- oVT__ In(Sp/K) + (= ry — 07 /2)Ta oVT
Both the domestic interest rate, r, and the foreign interest rate, r;, are the rates for maturity T. Putand call options on a currency are symmetrical in that a put option to sell X, units of currency Afor Xp units of currency B is the same as a call option to buy Xp units of currency B for X, units ofcurrency A.
=d,—oVT
Example 13.4 Consider a four-month European call option on the British pound. Suppose thatthe current exchange rate is 1.6000, the strike price is 1.6000, the risk-free interest rate in the UnitedStates is 8% per annum, the risk-free interest rate in Britain is 11% per annum, and the option price278 CHAPTER 13
is 4.3 cents. In this case Sy = 1.6, K = 1.6, r = 0.08, r- = 0.11, T = 4/12, and ¢ = 0.043. The impliedvolatility can be calculated iteratively. A volatility of 20% gives an option price of 0.0639, a volatilityof 10% gives an option price of 0.0285, and so on. The implied volatility is 14.1%.
From equation (3.13), the forward rate, Fy, for a maturity T is given by
Fy = Spe"
This enables equations (13.9) and (13.10) to be simplified to
 ¢ =e [FyN(d,) — KN()] . (13.11)p =e" "[KN(—d2) — FyN(—d,)] (13.12)where ,4, — FoR) +0°T/2i= 6 fFIn(Fy/K) — 0° T/2d= eee =d,— oVT
Note that, for equations (13.11) and (13.12) to apply, the maturities of the forward contract andthe option must be the same.
In some circumstances it is optimal to exercise American currency options prior to maturity.Thus American currency options are worth more than their European counterparts. In general,call options on high-interest currencies and put options on low-interest currencies ate the mostlikely to be exercised prior to maturity. This is because a high-interest-rate currency is expected todepreciate relative to the U.S. dollar, and a low-interest-rate currency is expected to appreciaterelative to the U.S. dollar. Unfortunately, analytic formulas do not exist for the evaluation ofAmerican. currency options. We discuss numerical procedures and analytic approximations inChanter 18
13.5 FUTURES OPTIONS
 
 
_
Options on futures contracts, or futures options, are now traded on many different exchanges.They are American-style options and require the delivery of an underlying futures contract whenexercised. If a call futures option is exercised, the holder acquires a long position in the underlyingfutures contract plus a cash amount equal to the most recent settlement futures price minus thestrike price. If a put futures option is exercised, the holder acquires a short position in theunderlying futures contract plus a cash amount equal to the strike price minus the most recentsettlement futures price. As the following examples show, the effective payoff from a call futuresoption is the futures price at the time of exercise less the strike price; the effective payoff from a putfutures option is the strike price less the futures price at the time of exercise.
Example 13.5 Suppose it is August 15 and an investor has one September futures call optioncontract on copper with a strike price of 70 cents per pound. One futures contract is on 25,000 poundsof copper. Suppose that the futures price of copper for delivery in September is currently 81 cents,Options on Stock Indices, Currencies, and Futures 279
and at the close of trading on August 14 (the last settlement) it was 80 cents. If the option isexercised, the investor receives a cash amount of
25,000 x (80 — 70) cents = $2,500
plus a long position in a futures contract to buy 25,000 pounds of copper in September. If desired,the position in the futures contract can be closed out immediately. This would leave the investor withthe $2,500 cash payoff plus an amount
25,000 « (81 — 80) cents — $250
reflecting the change in the futures price since the last settlement. The total payoff from exercising theoption on August 15 is $2,750, which equals 25,000" — K), where F is the futures price at the timeof exercise and-K-is the strike price.
Example 13.6 An investor has one December futures put option on corn with a strike price of200 cents per bushel. One futures contract is on 5,000 bushels of corn. Suppose that the currentfutures price of corn for delivery in December is 180, and the most recent settlement price is 179 cents.If the option is exercised, the investor receives a cash amount of
* 5,000 x (200 — 179) cents = $1,050
plus a short position in a futures contract to sell 5,000 bushels of corn in December. If desired, theposition in the futures contract can be closed out. This would leave the investor with the $1,050 cashpayoff minus an amount
5,000 x (180 — 179) cents = $50
reflecting the change in the futures price since the last settlement. The net payoff from exercise is$1,000, which equals 5,000(K — F), where F is the futures price at the time of exercise and K is thestrike price.
Quotes
As mentioned earlier, most futures options are American. They are referred to by the month inwhich the underlying futures contract matures—not by the expiration month of the option. Thematurity date of a futures option contract is usually on, or a few days before, the earliest deliverydate of the underiying futures contract. For exampie, the S&P 500 index futures options expireon the same day as the underlying futures contract; the CME currency futures options expire twobusiness days prior to the expiration of the futures contract; the C