FEDERAL RESERVE BANK OF ST. LOUIS JANUARY 1984

Are Options on Treasury BondFutures Priced Efficiently?Michael T. Belongia and Thomas H. Gregory

NTIL recently, trading in commodity options hasbeen viewed with a great deal of suspicion in theUnited States by both the general public and marketregulators. The low margin required by option markets

has led mans’ people to believe that unsophisticatedinvestors with limited resources were being encour-aged to speculate and that commodity price move-ments could be manipulated by sophisticated specu-lators using a high degree of leverage.1 Few peoplerealized the useful role that speculators in futures andoptions markets play in assuming risk that others de-sire to avoid (thus providing hedging opportunities)and providing better estimates of future spot prices.2

The Commodity Futures Trading CommissionICVFC) is gradually lifting restrictions on option trad-ing by allowing each commodity exchange to opentrading in options on one of its futures contracts. Thefirst phase of the CFTC pilot program introduced in1982 saw eight commodity exchanges participate by

offering options on several different futures contracts;these contracts covered three different stock marketindices, two weights of gold, heating oil, sugar and U.S.Treasury bonds.3~’l’hisarticle focuses on the pricing ofoptions on Treasury bond futures.

Michael T. Be/ongia is an economist and Thomas H. Gregory is asenior analyst at the Federal Reserve Bank of St. Louis.‘A recent overview of problems associated with options trading in theearly 1900s is provided in Wall (1983).

2One notable exception to this was Holbrook Working, who wroteextensively on the potentially useful role ot speculators. The in-terested reader is referred to Working (1977).

3For more detail on the specifics of the CFTC pilot programs and ageneral background to options trading, see Wolf (1982): and Belon-gia (1983).

The behavior of this particular option price series isinteresting for at least two reasons. First, if the optionsmarket is efficient, no arbitrage opportunities will existbetween any two option contracts.4 Stated differently,an efficient options market is one in which the samemarket price will be observed for options with the samelevel of risk and rate of return. Because efficiency is onecritenon that the CF’I’C is likely to consider when de-

ciding the future of this market, it is important to assesswhether the options niarket in U.S. Treasury bond fu-tures contracts satisfies this criterion.

The second motivating interest of this study is theusefulness of Black’s theoretical model in estimatingthe prices of American-type options on futures.5 Amer-ican options permit the holder to exercise the option atany time before the option contract expires. Most op-tion pricing formulas, however, attempt to explain theprices of European options, which can he exercisedonly on the expiration date of the option contract.

Although the Black model is widely accepted as atheoretical representation of option prce determina-

tion, some recent studies using stock options suggestthat its predecessor, the Black-Scholes model, does notfit market data well.6 Limited applications of the Black

4Etticient markets are those that reflect all available information.Weak form market efficiency implies that all information contained inpast price movements are fully reflected in current prices. Semi-strong efficiency suggests that current prices reflect all publicly avail-able information. Strong form efficiency means that prices reflect allinformation, both public and private. A considerable body of empiricalwork suggests that heavily traded capital markets are at least semi-strong efficient. See Fama (1970).

5Black (1976).6See, for example, Black and Scholes (1972); Gulteken, Rogalski andTinic (1982); Finnerty (1978); Whaley (1982); and O’Brien and Ken-nedy (1982).

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FEDERAL RESERVE BANK OF ST. LOUIS JANUARY 1984

model to the pricing of London commodity optionshave produced contradictory results about marketefficiency and the model’s applicability.’ In view ofthese results and the recent availability of options datafrom U.S. markets, it is of some interest to determinewhether the Black model accurately describes the pro-cess by which prices on U.S. Treasury bond options aredetermined. From a different perspective, the researchquestion is whether judgments about the observedbehavior of option prices can be based on comparisonsto prices predicted by this theoretical model.

This article first describes some basic principles of

options contracts and their relationship to futuresmarkets. The behavior of prices in the Treasury bondoptions market then is examined using atest proposed

by Latane’ and Rendleman.8

OPTIONS AND FUTURES IN THECVFC PILOT PROGRAM

Options trading may be clarified somewhat by firstcomparing it with futures trading. A futures contractobligates the holder to buy or sell) a specific volume ofthe underlying commodity at a specified price at somefuture date. An agreement to buy the commodity is a“long” futures position; a “short” position is an agree-ment to sell. Iffutures prices rise, holder’s oflong posi-tions realize a profit that is exactly offset by the losses of

the holders of short positions that day, and vice-versa.Futures contracts are settled each day with debits orcredits to the margin accounts of individuals holding afutures position. For example, if an individual bought aTreasury bond futures contract and, by the end of thatday, Treasury bond futures “settled” at a higher’ pnce,he would realize a profit equal to the change in thevalue of the futures contract less transaction costs. Hethen would have the choice of liquidating the futurescontract or holding it in hope of further price apprecia-tion.

Futures contracts normally call for delivery of a

homogeneous, standardized product. ‘rhe delivery ofhomogeneous, standardized Treasury bonds is com-plicated by the fact that Treasury bond prices respondto factors such as coupon rates and callability featuresthat are specific to individual issues of Treasury bonds.Thus, the Treasury bond futures contract, as specified

‘Studies of London options include Hoag (1982); and Figlewski andFitzgerald (1982).

°Latane’and Rendleman (1976).

by the Chicago Board of Trade, calls for delivery of ahypothetical 8 percent coupon Treasury bond not call-

able for at least 15 years from the date of delivery. Ifnocall provision is present, the bond must not mature for’at least 15 years from date of delivery.9 These bondshave a face value of $100,000 at maturity. A pr-ice of 70

implies a contract valued at $70,000.

An option contract gives its purchaser the right, butnot the obligation, to buy or sell a specified volume ofacommodity for a set price at some future time. Withinthe CFTC pilot program, this right to buy or sell appliesonly to specific futures contracts and not to the physi-

cal commodities underiying those contracts. For ex-ample, the purchaser of a call option on Treasury bondfutures buys the right to purchase a specific Treasurybond futures contract for a specified price prior to

some agreed-upon future date.

If, before that date, the market price of that Treasurybond futures contract rises above a specific level (thesum of the exercise price, the price of the call optionand any commission costs), the purchaser will find it

profitable to exercise the rights of the call option. Bydoing so, he buys the futures contract (that is, holds along position in the Treasury bond futures market) andobtains an immediate profit equal to the difference

between what he paid for the futures c~ontract(theexercise price of the call option) and the current mar-

ket price, less the transaction costs.

The purchaser of a put option, conversely. pur-chases the tight to sell aparticular’ futures contract at aset price. In this case, if the futures price falls below aparticular level, the purchaser will find it profitable toexercise the rights of the put option arid, by doing so.enter into a short position in the futures mar’ket. ‘rhiswill enable the individual to sell futures contracts forTreasury bonds at a price above the ctu’rent marketprice.10 In practice, owners ofboth call and put optionsoften choose to realize profits by selling the option

9The CBT publishes tables of conversion factors that translate all ofthe deliverable Treasury bonds into 15 year, 8 percent couponbonds. The conversion factors for bonds with coupons less than 8percent are less than 1, and the factors for bonds with coupons over 8percent are greater than 1.

‘°Byselling the futures contract, the individual agrees to deliver aspecific amount of Treasury bonds at a specified price at the expira-tion of the contract. Again, the individual realizes an immediate profitequal to the difference between what he sold the futures contract for(the exercise price of the put option), and that trading day’s futuressettlement price, less transaction costs. He also is faced with thedecision to liquidate or hold further.

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FEDERAL RESERVE BANK OF ST. LOthS JANUARY 1984

instead of exercising its privileges and entering into a

futures market position.

The Commodity Option Contract

The key elements of a commodity option contract

are the strike (or exercise) price, the futures contract towhich the option applies and the premium. The pre-mium — the price of the option — is competitivelydetermined, whereas other elements of the option arepart of the contract itself An “in the money” call optionis one whose strike price (theprice at which the option

owner may exercise the rights of the option) is less thanthe current price of the futures contract that underliesthe option; a call option is “out of the money’ if itsstrike price is greater than the price of the futurescontract. The reverse is true for put options. For exam-ple, if the current futures price is at 75, call optionswhose strike prices are less than 75 and put optionswith strike prices greater than 75 are in the money. Calloptions with strike prices greater than 75 and putoptions with strike prices less than 75 are out of themoney.

WHAT SERVICES DO TREASURYBOND OPTIONS PROVIDE?

One useful role that option and futures contractsplay is to transfer the risk associated with adverse priceswings from hedgers to speculators. Consider, for’ ex-ample, the manager of a pension fund who expectedinterest rates to rise, lie could hedge against the risk ofcapital loss in the price of his bond holdings by sellingTreasury bond futures. If rates did rise, losses in hislong position (bond holdings) would be at least partial-ly offset by gains in his short position (futures con-tracts).

Because an option’s price changes in response to the

price of its underlying commodity or security, optionsalso can he used to hedge against risk. In fact, at thehear’t of the Black and Black-Scholes models is theassumption that a totally risk-fl-ce hedge can he con-structed using options and either futures (Black mod-el) or securities (Black—Scholes model).

How To Interpret Option Prices

Table 1, a reproduction of one day’s report on trad-ing in Treasury bond options. indicates that onSeptember 13, 1983, options could have been boughton futures contracts dated for’ delivery in December

I t~’ f -

~

1983, March 1984 and June 1984; no options had yetbeen written on the September 1984 futures contract.The data in the table’s first column show the strikeprices of available options, while columns 2—4 give thepremiums associated with call options at those strikeprices.

The data in the table show, for example, that calloptions on March 1984 Treasury bond futures hadbeen written with strike prices between 68 and 80; thefutures price on this date was 70-29/32. Therefore, thepremium on a call option with a strike price of 68 isexpected to be the highest premium since it offers the

option purchaser the right to buy Treasury bond fu-tures at a 2-29/32 discount to the cur-rent market price.The difference between this discount and the price ofthe call (4-22/64) represents the market’s evaluation ofthe potential for’ futirr’e price appreciation of this con-

tract.

‘rhe table also shows that call premiums fall as strikeprices increase. Higher’ strike prices offer’ the option

purchaser the right to buy Treasury bond futures at aprice above the current market price. A buyer wouldpurchase these options only if he expected futurespr’ices to incr’ease substantially above the option’sstrike price before the option’s expiration date. ‘thisnegative r-elationship between call option premiumsand strike pr-ices also is illustrated in figure 1.

The data in columns 5—7 of table I show the pre-nliurns on put options for’ the same strike pTStIes listed

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FEDERAL RESERVE BANK OF ST. LOUIS JANUARY 1984

F, p,eei,,,, a’s put optm’’’. ill, st,ike p’ice

= premise a, coil opus’ “itt, ‘‘‘ike p’ke A

in column 1. Because a put option gives the purchasertile r-ight to sell Treasury bond futures, put optionpremiums tend to increase with strike prces; that is,the right to sell at a higher’ pr-ice has a greater valuethan tile right to sell at a tower price. This relationship

is depicted by tile up\x’ard-sloping line in figure 1. Inthis arld other- r-espects, the properties of put optionsare the mir-r-or- image of properties associated with call

options.

USING THE BLACK MODEL TODERIVE CALL OPTION PREMIUMS

information needed to estimate the pr-ice of a particu-

lar option with the Black model.

The test of the Black-Scholes model suggested byLatane’ and Rendleman provides an interestingapproach to comparing theoretical and actual option

prices. ‘their- reasotling is that if tile market is pricingoptions and risk efficiently, then, given r, the sameestimate of a’ should apply to all options traded for’ agivenfutures contract on a particular day. For- examnple,all options offered on October 26, 1982, for- the Decem-ber 1983 futur-es contr-act should yield the same im-plied expectation of frrture retut’ns if the assumptionsthat underlie the Black model ar-c true. This r’esult

Coils holds because the same risk-free hedge can be con-structed over this interval by constructing a portfoliousing differ-entoptions on the same futures contr’act, ifmar’kets ar’e efficient.’3

The Latanc’ and Rendleman test of the t3lack-Scholesmodel for’ data on stocks and stock options also can heused to test the applicability of the Black model for’determining prices of options on futures contracts.”Their test involves the following steps. On a par’ticularday, observe data on a var’tet of diflbrent options onfutures contr-acts for the same commodity— for exam-pie, all of the data for’ options on U.S. Tr’easury bondfutures shown in table 1. Insert these data, a value for’ rand a starting value of a’ into the Black model arid

solve for a final value of a’ that minimizes the differ-—ences between actual and estimated call option prices.If the Black model is a correct representation of corn-

mnodity options pricing and if the mar-ket is pricingoptions efficiently, one would expect to find estimnatesof a’ that were nearly identical across all optionstr’aded that (lay for’ the same futitres contract.r Con—

‘l’he Black model can he written as:r 1

Ii) Pt. = e” {F”N)d, ) — X4N)d=)] (see insert).

The only two parameter’s of the model that are rIotdirectly observable ate r, the risk—free nominal interestr’ate, amid a’, the variance of expected future returns ofthe under-lying futur-es contract. The risk-fl-ce nominalinterest rate can he pr’oxied, however’, by the cur-r-erltmar-ket rate on Treasury bills with maturities near theexpir’ation dates of the var-ious futures contr-acts.’’ The

determinatiorl of an appropriate value for’ a’, the ex-pected variance of future retur-ns, is the last piece of

‘‘Black (1976).‘2Because Treasury bills are backed by the U.S. government, the risk

of default generally is considered to be zero,

“In the abstract to their 1973 article, Black and Scholes assert “(i)foptions are correctly priced in the market, it should not be possible tomake sure profits by creating portfolios of long and shod positions inoptions and their underlying stocks.” Their use of the term “correctlypriced” markets is synonymous with what we are calling efficientmarkets. Black’s model uses the underlying futures contracts inplace of the underlying stocks.

‘4A strict test of market efficiency would compare the yield on a safeasset with the yield on a portfolio of hedged options and futures withcontinuously changing hedge ratios, Our reasoning is, however,that if the Black model does not predict option prices well, either themodel is incorrectly specified or markets are inefficient. Therefore,in the absence of any sysfematic relationship between actual andimplied option prices, conclusions about market efficiency on thebasis of our “buy and hold” strategy are still valid.

‘tmWe are indebted to Fischer Black for emphasizing the implieddifferences among estimates of cc’ for the same contract andobservation dates.

Relationship Between Premiums on Put and Call Options

Premium

=

pp

Current Strike pricefuture, price

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FEDERAL RESERVE BANK OF ST. LOUIS JANUARY 1984

nearly identical, one can conclude either that the Blackmodel does not estimate option prices accurately)givemi the use of Treasury bill rates as proxies for- ri orthat this market does not price options efficiently.15

‘°Thisconclusion alsodependson several otherassumptions as well.Because the Black model is derived for application to Europeanoptions that do not have early exercise privileges, a debate hasdeveloped in the literature concerning what value, if any, can beattributed to the early exercise privilege of American-type options.

Based on the workof Robert Merton, who argued that early exerciseof stock options had no value unless dividends were involved, onemight conclude that this problem is irrelevant in a study of options oncommodity futures because dividends are not involved. Moreover,in practice, American options are almost never exercised beforeexpiration. The reason is that the option has two potential sources ofvalue: its immediate exercise value (if any) and its potential for priceappreciation in the future. Thus, an investor— in most cases — willbe able to realize a greater profit by selling the option instead ofexercising it. In efficient markets, if we exclude options on assetsthat pay dividends, American and European options should bepriced similarly. See Merton (1973).

versely, if the different estimates of a’ are not very

9

FEDERAL RESERVE BANK OF ST. LOUIS JANUARY1984

ESTIMATION AND RESULTS

Observations on Treasury bond options were takenat six dates between October 1982 and April 1983.17 Oneach of these six dates, data were gathered for activelytraded options with large open interest. In total, data

were gathered on 53 call options with differ-ent strikeprices or- futures contracts. On these same dates, in-terest rates were observed for- Treasury bills maturingnear the delivery dates of the various futures contracts;these values were used to represent risk-free rates ofreturn fri.’2

‘I’hese data and starting values for the unobservablevariance of expected future returns (a’) were used tofind values for dr and d,, the two points at which thecumulative normal density must be evaluated. Equa-tion I then was solved for an estimate of the call optionprice. By using different values of a’, the Black modelwas solved iteratively until avalue of a’ was found thatminimized the difference between actual and esti-mated option prices to within ±one cent. The values

of a’ that produced the minimum differ~encesfor the 53option contracts considered are reported in table 2.

The estimates of a’ in the fifth column of table 2, ingeneral, suggest that estimates of the implied variationof future returns differ numerically across options

written on the same futures contract on the same day.The spread between highest and lowest estimates of a’range from 0.014 for options on September futurest,-aded on February 23 to 0.110 for options on Junefutures traded on April 4. It is not clear’, however, that itis possible to test whether these estimates of a’ are

statistically different from one another. Unknown arethe mean of expected returns, the number’ of trader-sdetermining the mean and variance of returns, and theshape of the distribution itself. Judgmentally, however,it would appear’ that these estimated differences aresmall. In half of the cases examined, the spread is 0.026points or less. In economic terms, this result implies

“The dates, which were not randomly chosen, are: October 26,November23 and December 27, 1982; January 26, February 23and April 4, 1983.

‘°Thesame risk-free hedge over differenf periods (using differentfutures contracts), may imply a different risk-free interest rate if theterm structure of interest rates is not flat. That is, given a “normal”yield curve, the implied risk-free interest rate over a period of threemonths (the remaining duration of one option contract), should beless than the implied risk-free interest rate over a period of sixmonths (the remaining duration of another option on a differentfutures contract), observed on the same day. Three-month andsix-month Treasury bill rates were used to proxy the risk-free rate,depending on the remaining length of the option contract.

that, in one-half of the options examined, the range ofestimates on expected variation of future returns wasless than three basis points.

The last column of table 2 reports the e,v post profitthat could have been obtained — in the absence oftransaction costs and taxes — if the individual optionhad been held until expir-ation. That is, the dollarfigures listed show the change in the value of the op-tion between the observation date and the last day itwas tm-aded. Asthe data indicate, options purchased ona particular day and held until expiration all tended topm-oduce profits or losses, regardless of strike prices. Inother words, no apparent s,ystematic relationship be-tween realized profits and certain characteristics ofthese options is revealed by the profit data in the table.

‘I’he point with respect to judging market efficiency isthat nothing in available market data indicate, ev ante,that these options would perform as they did. That is,none of the results in table 2 indicates a consistent exante signal for profit opportunities, a result consistentwith an efficient market.

Testing the Model with DirectEstimates of Cr2

Another way to test the Black model mnight be to usehistorical price data to construct a proxy for the ex-pected future variance of returns on the futurescontract.19 Given this estimate of a’ and using theTreasury bill r’ate to proxy the risk-free mate, we canobtain an implied value of a call option. If the Blackmodel “predictions” represent the “efficient prices,”an investor should buy those options that the modelimplies are underpriced and sell options that the mod-

el implies are overpriced. The results of this test arereported in table 3.

These results do not yield any consistent arbitrag-able profit opportunities. There is no apparent patter’n

either to the implied value of a’ or to the differencesbetween the actual and implied call prices that, e~ante, would indicate profitable options. Ifan investorhad bought any of the options in our sample on Janu-ary 26, 1983, or any December 1982 call options onOctober 26, 1982, he would have earned a profit on thechange in option prices. Likewise, anyone who boughtMarch 1983 or-June 1983 call options on November 23,

‘°Hisforicalvalues for a-2 were determined by estimating the varianceof the log of the ratio of successive days futures contract prices, upto the date at which a particular observation was taken; thisvariance, when multiplied by 365, approximates an annualized rateof return.

10

FEDERAL RESERVE BANKOF ST. LOUIS JANUARY1984

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===I~ :~ ====== ppp=~t PP P ~ 1= =~~~=cP =~ ~ ==== ~= p = ====p\===== p ==== =~ \== p=== = /== = *= = =P=== pP == ==== p = === .. = ===>= === =p =Pp.=== ‘p ==~‘== ==== p PP ~ = = p.

pP. = “ ==( 4 ====== p ======== ,~= =p===== =p=== =====P ~= =p~= ~=p= <=4= P,~ p p === pP. ‘ ===p=p ~ = == 4 ‘= === p = =p===== == p. =‘=p ==== pp == P=====1

\ / = p Pp P pP== ~,

======,====:= = = =1 ~ ,p=p.===p= ~ Pp==== p = / pp = ‘~ p P== =3=’ p = p = = == ===== p== ~p== p == = == = ========== * ==p==P == =p =

4= ~= pp=======p p ~ =r ~=‘~ ==~=== p I~= ~ ~ =p’~ ~= ~ p <=

.======= p== P = p=j4===== = p===== = ~<= = =P==p=== ,=<p=== P===pp===p=pp =p== ~ =pP= = p p= === p ==~ =2= 4= ‘ ppPpP=== ==== = pp= P=p= ==P = p==p === ~ ==.= —< 4= == p p ~=I====~= =t== P p~= ~ = Pp

~ I==s =~=>=~e~= I4=~I~I~I~=~=~ ~ :~==~=~~‘=~=~iiP~c::. IL I~I= ===~ = = p p p p p p= Pp tSlRaw=== .~. = pp=pP ====p=== = = t4>= == = 4=~=(C ==== p 4==4=P p~= ~p =, \=pp7~ =P< p\ p==p=pp P pp Pp P p =p== p =p/ P 4 p =. ~ a

= ==z4=

=P==pç == \PPP~p4=9s ~

= 4= = p==p==== ‘ = ~ =p ===== ~ ~=P4=p P==ppp ===, Pp~==~ ====~y=~y~ç=====~=,~=p p. =~ = P p ~ ~ ~ $ =p ~ ~ Ppp =~o$si:pp=. ==p PPP=

p1P Pp p P=[ ~P p

4= ~=1J$~==P==p == ===

ppp=p P4= ~ =‘h== ~ : ~= ~ ~ =>~~= ~ pp I~== ===1M**== ~= =: ~M ~ P; ~

===& ==p= ~ P *W <= ~ ~==~ ~7*= I ~ ~= ~ = p < at~w~===== ~

~pp~p/P~/ =p=~iI~I:==~I1t===1= I==~~==4=& ;=:~4IA=~:. ~

=.p= p pp =~ =, =P~ P= p~P p =<. P~ p P / ~

p===== === ,=:P====~ = P.pP’~ . Pp $===P=p==== .. ~ p p P== =‘ ==/PPp

p P p =P===i L . ~ == p ~~~~**TS$pS =~= ~ = >=~ ===p=== = = ==Pp~=~~===p :~p = == . ~ . =P ~ ,.

P===pp =P.= ~ p ~=p = ~ ==P~P=ç p P p ~ P == p.

pp ~~=p ~ == ~= .=P p 4 7 p p.p Pp p. ~ ~

p p p pP ~ ‘P .YP .p=P p P p. .~= ,===‘=p p

:::~ ‘ ~ :?P:~=~ p P~ p=~~ ~:~~I:== z=:=:I p ~=I=~=

===pps.\ p ===pP~ P 1’ p==’ ~=== pp =~== \=p=/ p p~= ~ ~ p K =~:si~p p. Pp. P ~= pp.’ pP p p p P. P=p ~~=p = == ~ = ~Pp. ~ = P=Pp=p 7 p==PP p. = p;;=p.p. \ p p p==P ==;=p,= ~ = = =~==p , = ===,=Pp. = ~= p.==.== ==

p pP p 7~ Pp==p= pp =p===pp= p= \=P==p4=p= p p= ====== PPp=P==7===P 7===p~==p p=p ~ ~I~I= I: CPP =~PP= p =7

=7= == == p=,p====== =p= P7.\ \ pP. = ===‘===P 7= = p p== = , , 7 == ‘== Pp.=

:===2~/K~=,~==!:~p.=: p=== ~ ~ =

= ~= ;~= p = =====~== ~Cr~ ~4= = p P == = ===~ P= ;====7 P~= = = ; 7= = I = =;=\e==,=~~I~==~. . = ~ .. == p

=1= ~ IL=== PP PPpP==p= P~ p. :1: ~ ~ =~ P=p= p ~7~~

==~~==

= p .=P’p= ~P =P p , :: = ==,= Pp.P= ==, p ==\=P ==“= P ==:pPp=P=== P = p 7 P/~P ~ p =p= = p. p p ~= Pp

7

p ~:= = p P p==~ p p p ~P pp p pPP PP p <.P P pp P P p P 7 P

7p P

P p P = p = ~P ~= p. P p P p~ p P

p p~ P= 7= ~ P =~~7 ‘I=~ /== P ~ ~ p ~ p p

=P =p/ Ij7~ I =P ==p=P===1

:~~pP ~ Pp. 7~ P /p /~P PP

= p =>==== 2> p.==4= ~ p =\=p=p ;=: = ~c4=~Pp ~= = == =~= =7 1 = = > ~I~== == === =5= ===PpP== = = = \=pp =77 P= =4= p 11

FEDERAL RESERVE BANK OF ST. LOUIS JANUARY 1984

1982, or’ December 27, 1982, or any September 1983 calloptions on Febr-uary 23, 1983, or April 4, 1983, wouldhave incurr-ed losses. Some options that the modelimplied wer-e underpriced eventually rose in price;other’s, however, declined fumther’. Similarly~highervariance of expected retum-ns is associated with bothprofitable and non—profitable options; r-elatively lowerestimates of a’ yielded the same mixed mesimlts.

Additional evidence of mar-ket efficiency is shown bythe absence of any consistent relationship betweenstrike price and profit or- loss. Profits ar-c sometimesnegatively associated with strike prices (for- example,June 1983 options on January 26, 1983), while on otheroccasions losses are negatively associated with strikeprices fSeptember 1983 options on April 4,1983). Thus,generally no predictable e~ante patter-n between strikeprices and profits can be identified.

CONCLUSIONS

The trading of options on commodity futures hasbeen permitted onl recently in the United States.Because the success and futur’e of the CFTC’s pilot

program in options trading will depend, in part, onjudgments about pricing efficiency, it is of interest tocompare actual prices with those of a model whosefundamental assumption is that option pricing isefficient, In those instances where the Black modelestimates of option prices differed from observed mar-

ket values, we were unable to find consistent arhi-tragable profit oppor-tunities. Thus, we wem-e unable toreject the assumption that Treasury bond optionprices are “efficient” in the fundamental economicsense.

REFERENCES

Asay, Michael R. “A Note on the Design of Commodity OptionContracts,” TheJournalofFutures Markets (Spring 1982), pp.1—7.

Belongia, Michael T. “Commodity Options: A New Risk Manage-ment Tool for Agricultural Markets,” this Review (June/July 1983),pp. 5—15.

Black, Fischer. “The Pricing of Commodity Contracts,” Journal ofFinancial Economics (January/March 1976), pp. 167—79.

Black, Fischer, and Myron Scholes. “The Valuation of Option Con-tracts and a Test of Market Efficiency,” Journal of F/nance (May1972), pp. 399-418.

“The Pricing of Options and Corporate Liabilities,”Journa/ of Polit/cal Economy (May/June 1973), pp. 637—54.

Fama, Eugene. “Efficient Capital Markets: A Review of Theory andEmpirical Work,” Journal of Finance (March 1970), pp. 3B3—417.

Figlewski, Stephen, and M. Desmond Fitzgerald. “Options on Com-modity Futures: Recent Experience in the London Market,” inMenachem Brenner, ed., Option Pricing: Theory and Applications(Lexington Books, 1982), pp. 223—36.

Finnerty, J. “The Chicago Board Options Exchange and MarketEfficiency,” Journal of Financial and Quantitative Analysis (March1978), pp. 29—38.

Gulteken, N., R. Rogalski and S. Tinic. “Option Pricing Model Esti-mates: Some Empirical Results,” Financial Management (Spring1982), pp. 58—69.

Hoag, James W. “The Valuation of Commodity Options,” in Bren-ner, ed. Option Pricing: Theory and Applications, pp. 183-221.See Figlewski and Fitzgerald.

Latane’, Henry A., and Richard J. Rendleman, Jr. “Standard Devia-tions of Stock Price Ratios Implied in Option Prices,” Journal ofFinance (May 1976), pp. 369—SI.

Merton, R. C. “The Theory of Rational Option Pricing,” Bell Journalof Economics and Management Science (Spring 1973), pp. 141—83.

O’Brien, ThomasJ., and William F. Kennedy. “Simultaneous Optionand Stock Prices: Another Look at the Black-Scholes Model,” TheFinancial Review (November 1982), pp. 219—27.

Samuelson, Paul A. “ProofThat Properly Anticipated Prices Fluctu-ate Randomly,” Industrial Management Review(Spring 1965), pp.41-49.

Wall, Wendy. “Agricultural Options Regain Favor Amid Hopes TheyWill Aid Farmers,” Wall Street Journal (November14, 1983), p. 27.

Whaley, R. “Valuation of American Call Options on Dividend-PayingStocks: Empirical Tests,” Journal of Financial Economics (March1982), pp. 29—58.

Wolf, Avner. “Fundamentals of Commodity Options on Futures,”Journal of Futures Markets (Winter 1982), pp. 391—408.

Working, Holbrook. Anne Peck, ed., Selected Writings of HolbrookWorking (Board of Trade of the City of Chicago, 1977).

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