1989 the Box Spread Arbitrage Conditions Theory, Tests, And Investment StrategiesThe Box Spread Arbitrage Conditions Theory Tests and Investment Strategies

8/14/2019 1989 the Box Spread Arbitrage Conditions Theory, Tests, And Investment StrategiesThe Box Spread Arbitrage Conditions Theory Tests and Investment Strat

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The Box Spread ArbitrageConditions: Theory, Tests, andInvestment StrategiesAimee Gerbarg RonnStanford UniversityEhud I. RonnUniversity of Texas at Austin

Tbis paper develops and tests arbitrage bounds fora combination of tvo option spreadpositions knownas a box spread. Tbis strategy involves the simul-taneous use offour options and creates a positionthat is equivalent to riskless lending. The no-arbi-trage conditions are compared to existing arbitragebounds and are tested using Chicago Board OptionsExchange data.

The efficiency of option markets may be tested usingeither a particular option pricing model-such as theBlack-Scholes model which relies on specific distri-butional assumptions-or more general pricing rela-tions that may be derived solely from the absence ofarbitrage opportunities under arbitrary distributionalassumptions. This paper follows the latter approach: Anarbitrage condition for the "box spread" is derived,compared with previous arbitrageconditions for options,and tested using Chicago Board Options Exchange(CBOE) data.Earlier versions of this paper were presented at the June 1987 meeting of theWestern Finance Association and the November 1987 Center for Research inSecurity Prices Conference. This project was begun when the second authorwas on the faculty of the University of California, Berkeley. Financial supportprovided by the Berkeley Program in Finance is gratefully acknowledged. Theauthors acknowledge the helpful comments and suggestions of George Con-stantinides; MarkGarman;HerbJohnson; David Modest; MarkRubinstein; financeseminar participants at the University of California, Berkeley, the University ofUtah, and the University of Chicago; the editors of The Review of FinancialStudies; Michael Brennan; and Michael Gibbons. They also thank Aamir Sheikhfor computational assistance. The authors are solely responsible for any errorscontained in this article. Address reprint requests to Dr. E. I. Ronn, Dept. ofFinance, College and Graduate School of Business, University of Texas atAustin, Austin, TX 78712-1179.The Review of Financial Studies 1989 Volume 2, number 1, pp. 91-108? 1989 The Review of Financial Studies 0893-9454/89/$1.50


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The Review of Financial Studies/ v 2 n 1 1989

A significant consideration in testing the box spread arbitrage relationor boundary condition is that it is independent of the stock price. As aresult, the tests we present do not suffer from the problems of obtainingsimultaneous price data from the stock and options markets. We also over-come the problem of obtaining simultaneous prices in the options marketby using bid-ask quotes rather than transaction prices; thus, we assumethat purchases are made at the ask price and sales at the bid price. Thismakes our results conservative and has the advantage that it considerstransactions that were feasible.'This article is organized as follows. Section 1 presents the theoreticaldevelopment of the box spread's arbitrage condition and compares thisbound with existing boundary conditions. Section 2 contains the empiricalresults using CBOE option prices for January 2, 1981. Section 3 expandsthe empirical tests to seven additional trading days over the sample period1977-1984. Section 4, in turn, considers the economic profitability of boxspread lending over the eight trading days covered in this article. Finally,Section 5 summarizes the analysis.

1. The Box Spread Lower Boundary Condition1.1 Deriving the boundary from first principlesFor a given stock and expiration date, assume there exist call and putoptions with exercise prices K1and K2> K1.Let the price of the call optionwith exercise price K1be denoted C(K1), and define C(K2), P(K1), andP(K2) analogously.2Consider the following investment strategy. Purchase a bullish verticalcall spread: purchase C(K1) and write C(K2). Also purchase a bearish ver-tical put spread: buy P(K2) and write P(K1). This position is denoted a boxspread, and its total cost is given by C(K1) - C(K2) - P(K1) + P(K2)Equation (1) presents the no-arbitrage lower boundary of this position:

C(K1) - C(K2) - P(K1) + P(K2) ' (K2 - K1)exp (-r,t) (1)where t is the time to maturity of the options and r, the continuouslycompounded spot interest rate for maturity t.The payoff of the box spread is independent of the terminal stock price:The value at maturity is always K2 - K1. To see this, note that the payoffof the position at maturity is

I To demonstrate that the use of bid-ask quotes gives rise to conservative results, note that the market makercommits to transacting (at least) one contract at the bid-ask quotes. Moreover, in an examination of allCBOE bid-ask quotes and trades for 42 trading days during March and April 1985, Vijh (1986) found thatthe ratio of trades within the bid-ask spread to those outside the spread is better than 5.7:1. Finally, recallthat Roll (1984) has argued that the effective bid-ask spread is narrower than the explicit, or quoted,spread.2 The current section's theoretical analysis abstracts from transaction costs for ease of comparability to Merton(1973) and Cox and Rubinstein (1985).

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that condition (2) is violated. Consider selling short the,,RHS (write theAmerican call, buy the European call, both at K,) and buying the LHS (buythe American call, write the European call, both at K1). This yields a positiveamount immediately. If the American option with exercise price K2 is notprematurely exercised, the position may be held until maturity, when thecash flows will net out at zero. If the K2option is exercised, the K, optionmay be exercised to receive K2 - K1 immediately; this may be invested ina Treasury bill maturing at the European options' expiration date t. If theK2option were exercised at time t < t, the wealth at maturity satisfies (K2- K,)exp[rrt(t - ')] > K2 - K1,where K2 - K, is the maximal obligationon the European options at time t. An analogous argument establishes theresult, relation (3), for the put options.5The present value of the (uncertain) dividend stream can be obtainedby buying the stock, buying the European put, writing the European call,and borrowing the present value of the exercise price. In an arbitrage-freeenvironment, the costs of purchasing these dividends is identical whetheroptions with an exercise price of K2or K1are used. Thus:S - c(Kl) + p(Kl) - Klexp(-rtt) = S - c(K2) + p(K2) -K2exp(-rtt)

Upon rearranging,c(K1) - c(K2) + p(K2) - p(K1) = (K2 - K1)exp(-rtt) (4)

Adding inequalities (2) and (3) yieldsC(K1) - C(K2) + P(K2) - P(K1) ' c(Kl) - c(K2) + p(K2) - p(Kl)

Using relation (4) now implies the desired box spread lower bound givenin relation (1).

1.3 The box spread boundary under transaction costsExplicitly accounting for the bid-ask spread is a straightforward modifi-cation of relation (1). The result isO(Ki) - CG(K2) - PI(K1) + Pa(K2) ' (K2 - K1)exp(-rtt)

where superscripts a and b denote ask and bid prices, respectively.Taxes only make the option-based lending strategy more profitable.Whenever capital gains are taxed at advantageous rates relative to ordinaryincome, then a positive probability that neither written option will beexercised against the position within the period required to attain capital

5 An interesting alternative proof of relations (2) and (3) relies on the value of an American call option asequal to the sum of (1) an otherwise identical European option and (2) an American option to purchasethe stock with a strike price of K + c(K): that is, C(K) = c(K) + qK + c(K)]. Since the value of a calloption is a nonincreasing function of its exercise price, qK2 + c(K2)] c C4K1+ c(KD)]. But this immediatelyimplies that C(K2) - c(K2) ' C(K1) - c(K,). Similararguments orputs yield P(K1) - p(K,) c P(K2) -p(K1). We are indebted to the Review's anonymous referee for suggesting the usefulness of relations (2)and (3) as well as this alternative proof thereof.

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Box Spread Arbitrage Conditions

gains status implies that the position can potentially yield a riskless rateof return at capital gains rates.2. Empirical Results-January 2, 1981

Empirical tests were conducted using the Berkeley Options Data Base forCBOE prices, and Data Resources, Inc., for the term structure of risklessinterest rates. The analysis describes the arbitrage opportunities at twostages: detection of available opportunities, and subsequent implemen-tation (or acquisition) of these option positions. In Section 4 the analysisis extended to seven additional days in less detail. The empirical tests forthese seven dates are performed to give a historical perspective to theJanuary 2, 1981, results.2.1 Detection stageThe data were collected in the following manner. Each hourly interval ofthe trading day was exhaustively scanned for any occurrence of four bid-ask quotes corresponding to a box spread combination, for all stocks andmaturity dates. When four such quotes were detected in a given hourlyinterval, the latest such hourly bid-ask quote for each option was collectedfor subsequent analysis. The data were then sorted by month of expirationand by size of the implied riskless interest rate.6The implied continuously compounded riskless interest rate implicit inthe four bid-ask quotes, R, is given by

1 /K2 -K1R, -t log( (5)where V= G(K2) - CG(K2) - PI (K1) + Pa (K2) and t is the time to maturity(measured in years).The Options Clearing Corporation provides for delivery on the fifthbusiness day following the date when the exercise notice is tendered tothe Corporation. Since options may be exercised at the latest opportunity,the parameter tin Equation (5) equals the number of days to maturity plusseven (expressed as a fraction of a 365-day year).Consideration of transaction costs is an important element of empiricaltests of arbitrage boundaries. For market makers and brokerage firms, thebid-ask spread constitutes the sole transaction cost. For public traders, thecost also includes commissions on the purchase/writing of options andcosts incurred if and when any of the options are exercised. Moreover, inthe case of the box spread, each leg of the investment strategy incurs acost regardless of whether it is partially offsetting the payoff of another leg.Clearly, these costs vary across the different types of public traders.Consider the example of a K= 45 option selling at a premium of $1.50.

6 Certain flagrantviolations of arbitrage conditions-the (relatively few) instances of C(K1) < C(K2) or P(K1)> P(K2)-were deleted from the analysis.

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Table 1Quoted one-month box spread riskless lending opportunities in excess of Treasury bill rates,January 2, 1981 (number of days to maturity + 7 = 22; one-month riskiess rate: 12.73 percent)Time of option transactions Ratio of Daily

Ticker Exercise prices (Central Time) tob number ofsymbol High Low First Last P, (in %) contractsAVP $ 35 $ 30 124537 125839 .00 0TDX 200 190 134714 135904 .11 53HM 70 60 150636 150948 .00 435TDX 220 190 135125 135956 .06 53TDX 220 200 150504 151117 .00 143HM 60 50 94146 95832 .38 59IBM 65 55 151123 151712 .00 80IBM 70 55 151123 151717 .00 80IBM 65 50 151120 151712 .00 20DD 40 35 151004 151131 .00 0HM 70 50 94146 95936 .38 59IBM 70 50 151120 151717 .00 20KMG 80 70 133200 135712 .79 149ARC 60 50 94201 95842 .39 0IBM 60 55 151123 151714 .00 80IBM 65 45 151116 151712 .00 0IBM 70 45 151116 151717 .00 0MER 35 15 111912 113606 .00 0TDY 170 150 95033 95747 .23 13IBM 75 55 151123 151719 .00 80IBM 75 50 151120 151719 .00 20Average

Variables are defined as follows: Pb is the high stock price in the interval of columns 4 and 5, P, is the lowstock price in the interval of columns 4 and 5, and NT indicates that at least one of four options is nottraded in last trading day.Daily minimum (column 7) is the minimum number of option contracts traded on January 2, 1981, takenacross all four option contracts; zeros indicate that the bid-ask quotes did not result in a trade.

Using a mid-1986 commission schedule of the Charles Schwab discountbrokerage firm, the commission on the purchase of 20 such option contractswould have been $62, and the commission for the purchase or sale of stockinvolved in the exercise of the options would have been $224. Because ofthe possible exercise of all four options in the box spread, this total costof $286 may be incurred as many as four times, with the resulting $1144constituting 127 basis points of the total exercise price of 45 x 20 contractsx 100 shares/contract = $90,000. For full-commission brokers, the anal-ogous costs would rise to 208 basis points. Large institutional traders, onthe other hand, have commission costs on the order of 5 cents/share, or20 cents per box spread.

The interest rates yielded by Equation (5) were examined under sixalternative transaction cost assumptions:* Zero commission costs.* Flat per-box spread transaction costs of 5 cents and 30 cents;7 for theflat fee cost of c, that interest rate is

7 Thus, if K2- K, = 5 and the zero transaction cost price is, say, $4.85, then the transaction cost would raisethe cost to $4.90 and $5.15, respectively. (In the latter case, the implied interest rate would be negative.)

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Table 1ExtendedImplied interest rates under alternative transaction cost scenariosZero 20 basis Liquidation of position atcost 5 cents 30 cents points 2.0% Noon Close85.1 67.7 -16.5 60.8 -112.3 24.3 24.385.1 76.4 33.5 16.7 -473.1 10.4 32.972.9 64.3 21.7 48.8 -123.1 10.4 10.457.4 54.5 40.3 32.4 -147.0 3.4 18.857.4 53.1 31.8 20.0 -254.8 -12.1 10.342.0 33.5 -8.3 21.7 -120.0 NT NT40.3 31.8 -9.9 18.4 -136.9 10.2 20.328.4 22.8 -5.0 12.8 -90.9 1.7 -27.226.7 21.1 -6.6 12.2 -82.0 6.7 13.423.4 6.6 -74.6 -3.3 -196.9 -125.8 -129.421.7 17.5 -3.3 10.0 -60.3 NT NT21.7 17.5 -3.3 10.0 -60.3 1.7 -20.321.7 13.3 -28.0 -5.0 -198.4 -61.6 -61.620.0 11.7 -29.6 0.0 -139.9 -75.6 -10.120.0 3.3 -77.8 -19.8 -313.5 -44.2 -44.220.0 15.8 -5.0 9.2 -53.9 5.0 10.016.7 13.3 -3.3 7.3 -42.6 0.0 -16.816.7 12.5 -8.3 10.8 -8.3 -10.1 -25.316.7 12.5 -8.3 -11.6 -217.4 -10.1 -30.415.0 10.8 -9.9 2.5 -74.6 -16.8 -27.013.3 10.0 -6.6 3.3 -52.3 -11.7 -20.234.4 27.2 -8.4 12.2 -140.9 -15.5 -14.3

1Rt - logI 12-K (6)t \V+ c/* Proportional commission costs of 20 basis points and 2 percentage

points; for a conservatively low estimate of the implied lending opportu-nity, these commission costs will be applied to the higher exercise price,K2.For proportional costs of -y,these rates are

t V+ -K\(2)* Liquidating the position, by selling the box spread on the last tradingday; this strategy eliminates the need to exercise an option but is exposed

to the price risk of the option at maturity. Two values of box spread liq-uidation are calculated: the values at noon and at close of the last tradingday. For a maturity liquidation value of L, the interest rate isR = log(j) (8)

Tables 1 to 3 present, for January 2, 1981, the one-month, four-month,97


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Table 2Quoted four-month box spread riskiess lending opportunities in excess of Treasury bill rates,January 2, 1981 (number of days to maturity + 7 = 113; four-month riskiess rate: 15.29 percent)Time of option transactions Ratio of Daily

Ticker Exercise prices (Central Time) (Pt - P,) m bnmumsymbol High Low First Last P, (in %) contractsTDX $200 $190 150141 151057 .00 1TDY 180 170 144441 144539 .00 1HOI 50 40 141136 145840 .22 4HOI 50 35 94805 95916 .46 0ARC 60 50 151244 151347 .00 17HOI 50 25 140920 144730 .00 0STK 25 20 94637 95720 .57 6ARC 60 45 94215 95824 .39 0HM 70 50 131450 135726 .18 48ARC 60 40 151236 151347 .00 0BNI 60 50 142908 145355 .55 0HOI 60 25 140920 144208 .24 0HOI 60 35 95243 95917 .22 0HOI 60 40 141136 145840 .46 0HOI 50 45 141139 145749 .24 12HOI 45 25 140920 145749 .24 0FLR 50 40 131606 131916 .00 0ARC 70 40 151236 151350 .00 0ARC 70 45 94215 95807 .18 0HOI 50 30 94805 95916 .46 0ARC 70 50 151244 151350 .00 5HM 70 60 132211 134555 .00 48BNI 70 50 142908 145723 .55 0KMG 80 70 133136 134840 .47 0MER 35 25 143451 144629 .66 0MER 40 25 152227 152309 .00 0MER 40 30 152229 152309 .00 0HOI 60 30 95243 95917 .22 0AverageVariables are defined as follows: Pb is the high stock price in the interval of columns 4 and 5, P, is the lowstock price in the interval of columns 4 and 5, and NT indicates that at least one of four options is nottraded in last trading day.Daily minimum (column 7) is the minimum number of option contracts traded on January 2, 1981, takenacross all four option contracts; zeros indicate that the bid-ask quotes did not result in a trade.

and (the maximal length) nine-month investment opportunities yieldingimplied detection-stage interest rates in excess of the relevant Treasurybill spot interest rates.It is apparent from the tables that riskless interest rates in excess of the8 For several box spreads, the quoted times exceed 151500. Though infrequent, there are instances wherethe time-stamping of the quote was delayed beyond 3:15 P.M. In most cases, the time lag between thequote and its time-stamping does not exceed 40 seconds. Also, since several box spread opportunitiesappeared at multiple hour intervals, only the highest implied interest rate for each spread is presented inthe tables which follow.9 Columns 6 of Tables 1a to 1 c present the (percentage) movement of the underlying stock price betweenthe time of detection of the first and fourth option contracts constituting the box spread: (Pb -P,)P,,where Pb (P,) represents the high (low) stock price during the interval of detection. The low magnitudesappearing in these columns show little, if any, stock price movement in this interval, demonstrating thatthe investment opportunities available therein are unlikely to be attributable to a movement in the priceof the underlying stock.

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Table 2Extended

Implied interest rates under alternative transaction cost scenariosZero 20 basis Liquidation of position atcost 5 cents 30 cents points 2.0% Noon Close76.9 74.9 64.9 61.0 -50.2 NT NT38.7 36.9 28.0 25.9 -66.1 NT NT31.5 29.8 21.0 28.0 4.2 NT NT31.2 30.0 24.1 28.8 15.0 NT NT29.7 28.0 19.3 25.5 -3.9 11.1 23.929.7 29.0 25.5 28.3 22.7 NT NT29.7 26.2 9.2 26.2 2.6 25.9 17.226.6 25.4 19.6 23.8 6.2 10.0 24.025.5 24.7 20.3 23.1 8.5 19.3 19.325.2 24.3 20.0 23.1 11.5 13.0 23.225.2 23.4 14.9 21.0 -8.0 15.0 15.024.8 24.3 21.9 23.6 19.9 NT NT23.8 23.1 19.6 22.1 14.2 NT NT22.0 21.2 16.9 20.0 8.5 NT NT20.7 17.2 0.6 13.9 -36.0 -4.2 -4.219.6 18.8 14.5 18.1 11.2 NT NT18.6 16.9 8.5 15.2 -7.7 14.7 14.718.3 17.7 14.9 16.7 9.3 1.7 17.017.6 16.9 13.5 15.7 5.5 -2.4 16.017.6 16.7 12.5 15.9 7.5 NT NT16.9 16.1 11.8 14.5 0.3 -5.1 14.016.9 15.2 6.9 12.2 -21.6 6.7 -1.716.6 15.7 11.5 14.2 0.0 13.3 13.316.6 14.9 6.5 11.2 -27.8 -4.1 4.416.6 14.9 6.5 14.2 0.0 8.7 6.416.6 15.4 9.8 14.8 5.4 3.0 9.716.6 14.9 6.5 13.9 -3.2 4.4 6.415.9 15.3 12.5 14.5 9.2 NT NT24.5 23.1 16.5 20.9 -2.2 7.7 12.9

Treasury bill rate may be available in option markets.9 These rates areavailable primarily to market participants able to transact at low commissioncosts, but not to public traders who incur higher commission costs.10 Fur-ther, market depth, or liquidity, is available for implementation purposesprimarily in the near- to midterm maturities (up to four months). Finally,note that the effect of time to maturity on the desirability of option positionsis ambiguous. For zero-transaction costs agents, market depth and thesupernormal available interest rates in excess of Treasury bill rates taperoff as time to maturity increases. The inclusion of transaction costs increasesthe relative attraction of intermediate-term contracts (two- to four-monthmaturities) since the transaction costs are amortized over a longer period

10 Moreover, these rates are obtainable predominantly through the exercise of all in-the-money options atmaturity. Due (presumably) to taxes and transaction costs, the prices of options are inferior to their intrinsicvalues at maturity, and thus the liquidation of the position prior to maturity (columns 13 and 14) doesnot appear to be a profitable alternative.

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Table 3Quoted nine-month box spread riskless lending opportunities in excess of Treasury bill rates,January 2, 1981 (number of days to maturity + 7 = 267; nine-month riskless rate: 14.40 percent)Time of option transactions Ratio of Daily

Ticker Exercise prices (Central Time) (Pb-oP) mbnimumsymbol High Low First Last P, (in %) contractsKN $25 $20 125708 125744 .00 0FT 70 50 151031 151111 .00 0FT 70 60 95109 95137 .00 0LIT 80 70 123814 123821 .00 0AverageVariables are defined as follows: Pb is the high stock price in the interval of columns 4 and 5, P, is the lowstock price in the interval of columns 4 and 5, and NT indicates that at least one of four options is nottraded in last trading day.Daily minimum (column 7) is the minimum number of option contracts traded on January 2, 1981, takenacross all four option contracts; zeros indicate that the bid-ask quotes did not result in a trade.

of time; conversely, liquidity becomes more restricted in the longer-matu-rity options.

2.2 Implementation stageThe detection of superior lending opportunities in the options markets isno guarantee of their subsequent availability for implementation purposes.Market prices may display adverse price movements that make previouslydesirable positions disadvantageous. Moreover, the computer algorithmselected to detect profitable lending opportunities will perforce select themost advantageous positions. Such a methodology begs the question ofdetection-stage data errors, that is, possibly erroneous price quotationswhich led to seemingly large rates of lending in excess of Treasury billrates. A careful scanning of the data revealed that a substantial number ofpositions profitable at detection showed no subsequent bid-ask quotes ortrades in the remainder of the trading day. To a certain degree, this over-states the number of positions that may be attributable to possible "dataerrors." Many profitable positions were detected toward the end of thetrading day: The lack of subsequent implementation opportunities may beattributable to the termination of the day's trading and does not necessarilyimply that the seemingly profitable positions constituted data errors. Never-theless, the implementation analysis below will focus only on those optionsin which a subsequent bid-ask quote appeared at some time during theremainder of the trading day.Implementation of these investment strategies hinges on two elements:the implementation procedure, that is, the sequence by which the fouroptions are acquired; and the lag time between detection and implemen-tation.

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Table 3Extended

Implied interest rates under alternative transaction cost scenariosZero 20 basis Liquidation of position atcost 5 cents 30 cents points 2.0% Noon Close18.1 16.5 9.0 16.5 6.2 NT NT15.5 15.1 13.2 14.4 8.0 NT NT14.4 13.6 9.9 12.3 -2.7 NT NT14.4 13.6 9.9 12.0 -5.4 8.4 2.915.6 14.7 10.5 13.8 1.5 8.4 2.9

Consider first the sequence of option acquisition. In practice, imple-mentation of this position requires the purchase of two option spreads.However, in simulating the acquisition process from the tapes, one mustspecify the process by which the position is established. The first methodwe consider simply seeks the four bid-ask quotes irrespective of the orderin which they appear on the tapes. That is, after an exogenously specifiedlag time has been imposed, the first four relevant bid-ask quotes appearingon the tapes (in the chronological order in which these were offered tothe market) are used to establish the position. This procedure assumesthat the two option spreads could be established at the then-current bid-ask spread and is denoted the "unconstrained" methodology, since it doesnot specify a particular ordering for the purchase of the requisite position.The second method is less exposed to execution risk and is referred toas "conservative": The position is established so that the purchaser is notexposed to unlimited risk. This method requires that no option be writtenwithout its counterbalancing option having been purchased. The (six)resulting implementation paths guarantee the limited risk property becausethe lower (higher) striking price call (put) is purchased prior to the writingof the counterbalancing call (put).A key ingredient of the implementation stage analysis consists of anexogenously specified lag time between the detection of aprofitable (option-based) lending opportunity and its implementation, or acquisition. This"recognition lag" is designed to proxy for the time interval required foran investor to receive the data, analyze it, and proceed to execute aninvestment strategy. As such lag times will differ across alternative classesof investors, the subsequent analysis will consider a range of six recognitionlag times: (1) implementation in the four quotes immediately followingdetection; (2) through (5) implementation after 5 minutes, 10 minutes,30 minutes, and 1 hour; (6) for trades detected after 3 P.M., implementationat first quotes on the succeeding business day.The top part of Table 4 summarizes the January 2, 1981, implementation

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stage analysis for lending opportunities.11 The diminished rates of returnavailable 10 minutes after detection constitute evidence in favor of CBOEefficiency in rapidly revaluing the prices of temporarily mispriced securi-ties. The result that prices adjust within 10 minutes is stronger than thatof Klemkosky and Resnick (1980), who found that profit opportunities stillexisted after 15 minutes. Klemkosky and Resnick used both stock andoption markets in their analysis, so that their strategies are subject tointermarket nonsynchroneity problems as well as to the possibly greaterdifficulty of implementing intermarket, rather than intramarket, arbitragestrategies. Moreover, they use transaction prices, and their failure to takecognizance of the bid-ask spread results in apparently more advantageousand longer-lasting profit opportunities.An additional implementation constraint may be imposed. Consider theunconstrained and conservative strategies with the additional restrictionthat the position be fully implemented (i.e., all four positions be acquired)within a span of 10 minutes following the exogenously specified detection-implementation lag time. This procedure mimics most accurately the CBOEterminology of "fill-or-kill" order: That is, after the exogenously specifiedlag time (of alternative length of 5, 10, ..., 60 minutes), the position iseither "filled" within 10 minutes or "killed."'12The lower half of Table 4 presents the empirical results of such astrategy. Once again we observe that the (less numerous) profitable oppor-tunities are eliminated if the acquisition procedure is initiated in an intervalthat exceeds 10 minutes from the point of detection.The issue of incompletely implementable box spread positions, for whichthe bid-ask quote on one (or more) of the four underlying options is notavailable in the remainder of the trading day, may be investigated by esti-mating the cost of "legging out" of the partially completed box spread.Let the execution time, from initiation of the execution procedure to theacquisition of the fourth option, be 10 minutes, and consider the imple-mentation procedure of the unconstrained form. Any box spread notacquired within 10 minutes of its initiation (after the exogenously specifiedrecognition lag) is deemed unimplementable for this exercise and itsreversal (legging out) is begun. Of the 475 "unimplementable" positionsdetected before 2:00 P.M. on January 2, 1981, only 31 were not fully revers-ible in the remainder of the trading day, and many of those were concen-trated in selected stocks. For the fully reversible positions, the net cost oflegging out ranged from $46 to $65 per $1000 positions.Further evidence on implementation opportunities is provided byinspection of rows 2 and 3 ("Detection" and "Immediate implementa-tion") of Table 5, which extends the intraday unconstrained implemen-

Inspection of Table 4 reveals that the "next business day" implementation yields fewer and significantlyworse investment opportunities. This is predominantly the case in the other tests performed below. Mostsubsequent results reported below will contain intraday values only.12 Note that market makers are precluded from issuing fill-or-kill orders.

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Table 4Average realized interest rates for one-month riskless lending onJanuary 2, 1981, for box spreadpositions with returns exceeding Treasury bill rate at detection (one-month riskless rate: 12.73percent)

ImplementationAnnual- on nextized Imme- business dayinterest diate Reonto a eoeRate atrate at imple- Recognition lag before Rate at imple-detec- menta- implementation detec- menta-Procedure tion tion 5 min 10 min 30 min 1 hour tion tion

Uncon- Average 37.2% 38.3% 39.0% 11.2% 13.8% 9.7% 28.5% -33.0%strained rateNo. of 18 18 18 16 16 14 13 13positions

Conser- Average 37.2% 35.9% 36.3% 13.8% 13.8% 9.4% 21.9% -49.4%vative rateNo. of 18 18 17 16 15 14 13 13positions

Execution interval not exceeding 10 minutesUncon- Average 37.8% 32.3% 45.5% -11.5% 5.0% -4.7% 28.5% -33.0%strained rate

No. of 8 8 7 5 8 5 13 13positionsConser- Average 45.4% 28.2% 45.7% -1.6% 15.9% -4.0% 57.4% 10.8%vative rate

No. of 5 5 4 4 5 4 1 1positions

Recognition lag before implementation refers to the exogenously imposed lag time between the detectionof a profitable box spread lending opportunity and the acquisition of the first (of the four) option contracts.Unconstrained strategy denotes acquisition of box spread using first four bid-ask quotes in chronologicalsequence. Conservative strategy denotes acquisition of box spread using bid-ask quotes such that acquireris never subject to unlimited risk. Average rate of interest is calculated by holding an equally weightedportfolio of all advantageously priced box spreads and then calculating the implied spot interest rate onthat portfolio. Execution interval not exceeding 10 minutes refers to acquisition of the four box spreadoptions in a time interval not exceeding 10 minutes from initiation to completion.

tation analysis to all maturity months. This table reveals that of the 158profitable positions available at "Detection" and "Immediate implemen-tation," only three were not subsequently available at "5-Minute lag";moreover, the average interest rate available at "Immediate implementa-tion" was essentially identical to that available at "Immediate implemen-tation" and at the "5-Minute lag." This suggests the following investmentstrategy: use "Immediate implementation" as confirmation of the interestrate available at "Detection stage" before submitting the required ordersfor acquisition of the box spread.Of all maturity months in Table 5, only maturities of three, five, and sixmonths display a relatively large number of box spread positions yieldinglending opportunities in excess of Treasury bill rates. Indeed, the resultsof Section 2.2 are predicated on the assumption that implementation of

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Table 5Extended

Months to maturity4 5 6 7 8Rate Rate Rate Rate Rate(%) No. (%) No. (%) No. (%) No. (%) No.15.29 - 14.56 - 14.55 - 14.45 - 14.53 -20.8 14 22.4 33 21.3 18 34.5 2 20.6 815.6 14 20.7 33 21.5 18 27.6 2 20.6 817.7 14 20.4 33 21.5 18 31.4 2 20.6 817.6 13 20.0 32 21.5 18 31.4 2 20.6 818.1 12 17.8 31 21.7 18 30.1 2 20.6 811.2 9 16.2 27 22.6 16 20.1 2 20.6 6

number of such positions was small, the profitability did not persist beyondthe detection stage, or both. Consequently, the maturing of the CBOEmarket and the increased volume appear to have generated some profitableinvestment opportunities in the intermediate time period. More recently,however, an increased efficiency of the market apparently eliminated most,if not all, arbitrage lending opportunities.154. An Assessment of Economic Profitability of Box Spread Lending

Sections 2.2 and 3 empirically demonstrate that, for low transaction-costagents, realizable opportunities for arbitrage profits existed, provided thatthe agents implemented their investments in an expeditious manner. It isimportant, however, to consider not only the realizable interest rate, butalso the magnitude of the arbitrage profits attainable through the use ofbox spreads.An examination of profitability was conducted as follows. Let njf be theminimum number of option contracts in box spread j (across the fouroptions constituting box spread j) which were actually traded at tradinghour r. From the analysis of Section 3, the most profitable opportunitieswould have resulted if agents could trade at the prices offered at thedetection stage. Then, assuming one could transact max{1, njr} number ofoption contracts at the bid-ask quotes, the profit per box spread is givenby the present value of the excess rate of return realizable on the dollarvolume existing in box spread j:

100(K2j - K1j)max{1, njr}[exp(-r,t) - exp(-Rjrt)]

15 In March 1983 the CBOE began trading in the OEX S&P 100 index options. The cash settlement featureof these contracts eliminates the transaction-cost issue involved in liquidating the box spreads. For theseoptions, which were examined separately, it was found that realizable profits were not economicallysignificant, the number of implementable positions was relatively small, and advantageous implementationrequired expeditious trading no later than a five-minute lag period.

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Table 6Quoted three-month box spread riskless lending opportunities in excess of Treasury bill ratesJuly 5, 1983 Feb. 1, 1984

March 1, 1978 Jan. 2, 1981 Dec. 30, 1982 Aver- Aver-Average Average Average age agerate Num- rate Num- rate Num- rate Num- rate Num-(%) ber (%) ber (%) ber (%) ber (%) berTreasurybill rate 6.56 - 14.92 - 8.15 - 9.29 - 9.10 -Detectionstage 11.1 1 22.8 45 22.9 82 16.5 29 18.0 11Immediateimplementation 11.1 1 19.8 45 12.6 82 8.2 29 5.9 115-Minute lag 11.1 1 19.8 43 15.3 82 7.6 29 6.3 1110-Minute lag 11.1 1 21.1 37 16.8 81 4.4 29 6.3 1130-Minute lag 5.1 1 20.2 36 15.3 78 3.0 29 6.6 111-Hour lag -10.8 1 20.9 35 9.5 71 4.8 28 0.6 11No profitable positions were detected on October 10, 1977, August 1, 1979, and May 1, 1980. Average rateof interest is calculated by holding an equally weighted portfolio of all advantageously priced box spreadsand then calculating the implied spot interest rate on that portfolio.

where RJT rt, Rjtis as given in Equation (5), rt is the Treasury bill ratefor maturity t, and the coefficient 100 follows from the recognition thateach option contract represents 100 shares of the underlying security. Thus,the daily profitability under these assumptions is obtained by summingacross all box spreads, at all hours r, and across all maturity months:169II 100 (K2, Kj)max{1, nfr}[exp(-rtt) - exp(-Rfrt)] (9)

Table 7 presents the calculation of each II for each the eight selecteddays in the data period 1976-1984.17 Based on the values of Table 5, thematuring of the CBOE seems to have generated two conflicting effects. Atfirst, a small number of option contracts resulted in low levels of "profit-ability."18The greater number of contracts over the years 1981-1983 allowedfor greater profitability, which was subsequently reduced, possibly due togreater market efficiency.Beyond the time-series pattern itself, it is important to comment on themagnitude of the dollar values. Recall that to achieve an arbitrage profitof $12,500 a day it was necessary to employ a constant monitoring of optionprices, to be able to transact quickly at low transaction costs, and to repro-16 The summation 19., in Equation (9) results from the recognition that the longest-maturity optionat any point in time is the nine-month option. Since only three maturity dates of options are traded at anypoint in time, the profit is identically zero for all maturities t for which options are not traded.

17 Due to the exhaustive nature of the profit calculation-that is, the summation over all profitable oppor-tunities T1-one bad quote can drive an arbitrage opportunity on multiple box spreads. In order to avoidthis double counting, each quote was used only once, in the position wherein it generated the greatestprofit.18 Of the $10,243 profit on October 10, 1977, $9045 resulted from the substantial volume in the options ofthe Eastman Kodak Company.

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Table 7Arbitrage profits, II, for selected datesDate Number of profitable positions Daily profit, IIOctober 10, 1977 13 $10,243March 1,1978 1 16August 1, 1979 3 38May 1, 1980 7 116January 2, 1981 149 9,719December 30, 1982 239 11,615July 5, 983 293 12,369February 1, 1984 79 2,904Daily profit, II, is the present value of the excess rate of return realizable on the dollar volume existingin a given box spread, summed across all box spreads, all trading hours, and all maturity months. [SeeEquation (9).]

duce the dollar transaction volume without affecting the bid-ask quotes.The confluence of these factors implies that the CBOE market, while per-haps not fully efficient at all times with respect to these arbitrage bounds,was in fact efficient in terms of economic significance.

5. SummaryThis article has derived the lower arbitrage bound on the option spreadposition known as the box spread. Tests of the bound using Chicago BoardOptions Exchange price data on listed options were carried out by usingthe riskless lending opportunities implicit in the box spread position acrossseveral trading days over an eight-year period. The results indicate thatarbitrage conditions in lending opportunities appear to exist only for lowtransaction-cost agents who can implement options trading expeditiously.Even under these conditions, the magnitude of these arbitrage profits maynot be economically meaningful on most trading days.The current article suggests several issues for future research. First, thereare a number of dynamic trading strategies implied by the model but notempirically tested. For example, an advantageously purchased lendingposition may be profitably sold if the postacquisition implied interest ratefalls below the then-current spot interest rate. Second, it may be instructiveto analyze the intra-implementation strategy of optimally "legging into" aprofitable position or "legging out" of a heretofore profitable but subse-quently unprofitable (or subsequently unimplementable) investment posi-tion. Finally, empirical testing of other arbitrage bounds may shed addi-tional light on the efficiency of options markets.

ReferencesCox, J. C., and M. Rubinstein, 1985, Options Markets, Prentice-Hall, Englewood Cliffs, NJ.Glosten, L. R., and P. R. Milgrom, 1985, "Bid, Ask and Transaction Prices in a Specialist Market withHeterogeneously Informed Traders," Journal of Financial Economics, 14, 71-100.

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Klemkosky, R. C., and B. G. Resnick, 1980, "An Ex-Ante Analysis of Put-Call Parity," Journal of FinancialEconomics, 8, 363-378.Merton, R. C., 1973, "Theory of Rational Option Pricing," BellJournal of Economics and ManagementScience, 4, 141-183.Roll, R., 1984, "A Simple Implicit Measure of the Effective Bid-Ask Spread in an Efficient Market,"Journalof Finance, 39, 1127-1139.Vijh, A. M., 1986, "Liquidity of Equity Options on the CBOE: Analysis of the Bid-Ask Spread, and the Priceand Information Effects of Trading Volume," working paper, University of California, Berkeley, November.

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1989 the Box Spread Arbitrage Conditions Theory, Tests, And Investment StrategiesThe Box Spread Arbitrage Conditions Theory Tests and Investment Strategies