21383,406378,352368,788355,152351,300381,265
FinancialInstruments2,758,5393,799,0043,749,8453,778,6144,052,5565,337 ,3525,372,623
Currencies317,970338,829299,178335,580413,150515,176477,527
15,197,631 185,397,113 28,808,45618,231,411 252,579,136 30,371,77417,393,317 259,024,379 24,293,46116,938,969 234,261,790 21,213,93817,093,481 250,143,412 25,210,99717,044,818  319 ,916,853  27 ,590 ,35017,294,322 303,684,764 23,558,563184,621189,502157,323132,507129,977163,804128,557
1,035,3751,809,4181,939,9081,903,9742,385,8862,705,7002,230,017
503,838578,209521,473502,451670,721865,921837,687
*Revised since FY 1998 annual report.Note: The number of contracts settled by delivery or cash settlement indicates the number of contracts actually expiring as opposed tothe positions traded and liquidated or rolled over before expiry.(Source: Commodity Futures Trading Commission, Annual Report 1999, www.cftc.org. Data for U.S. markets only.)
pay more for a futures contract than for the underlying cash product, and conversely. It follows that a futures contract has a priceequal to its underlying cash price on expiration day. This price isusually referred to as the special quotation (SQ) price.. Because cash corn is available for purchase at any time, the potential buyer of a futures contract has always an explicit choicebetween futures and cash. If he or she buys cash corn, he orshe receives delivery the same day, loses the bene®t of the cashhe or she disbursed, and has to store the corn if he or she doesnot have an immediate use for it. On the other hand, if he or shebuys the futures contract, he or she earns interests on the cashsaved until expiry and does not have any storage cost.Furthermore, he or she is still certain to receive corn but at afuture date that may be more suitable to his or her needs.Therefore, his or her decision to buy a futures contract will necessarily include these three components: interest rates, storage, and
114
THEORY AND TOOLS
convenience.4 But the problem has one more dimension becausecash corn will also be available at the expiry of the future. Thepotential buyer could do nothing, wait, and buy cash corn later,at a price unknown as of today with the risk that it will be moreexpensive than the cash price or the futures price today.Therefore, the buyer should enter a transaction today, eithercash or futures, only if he or she expects the price of corn to goup in the future. Now consider that he or she buys a future at $20per unit, and at expiry of the futures the cash corn sells for $15per unit. Because the futures is an obligation, the $5 potentialsavings is lost no matter what. The general conclusion is that afutures contract induces an exposure on the underlying cash price,despite the fact that no exchange of any sort has taken placebefore expiration date. Note that the $5 opportunity lost to thebuyer accrues to the seller. This is a zero-sum game, like allfutures transactions.. There is no payment of nominal between the transaction date andsettlement, as we mentioned.5 Suppose that a futures contract sellerloses its whole corn reserve in a ®re. He will be unable to delivercorn on the expiration date and will have to buy cash corn atmarket price because he has an obligation to deliver. Therefore,he carries an additional risk if he is not able to deliver for anyreason.6 This situation is described as a naked positionÐthat is, aderivative position without the protection of the underlying asset,as opposed to a covered position in which he would be able to delivercorn he owned. Our simple analysis shows that there is a criticaldifference between the two. A position in a future entered to offsetan opposite exposure on the underlying is a covered transaction, in
4
Financial futures have a different nomenclature. The convenience aspect is capturedby the monetary ¯ows expected during the life of the contract for a holder of theunderlying asset. In the case of equities these ¯ows are primarily dividends, includingthe tax rebate if there is one. Storage costs are the costs or bene®ts of holding aportfolio. For a short position, this cost is a borrowing fee; otherwise, it is the potentialrevenue generated by lending the stocks out. Note that storage costs are always anout¯ow for a commodity but can be of any sign with a ®nancial asset.5Futures listed on organized market have intermediate ¯ows as we will see, the margincalls. In our case the nominal payment of the corn underlying the contract is not madeuntil expiry.6It looks as though this risk is particular to the seller and that the buyer is immune to it.In fact, it is not exactly true because the buyer may need the corn before expiry, whichwould force him or her to buy it in the cash market. He or she would then, at theexpiration of the futures, have to sell the corn delivered back in the market.
CHAPTER 3 FUTURES AND RELATED INSTRUMENTS
115
which the portfolio is hedged and global risk decreased.7 A nakedfutures position gives rise only to a speculative risk.. Each party is exposed to the risk that its counterpart will fail tohonor its contract upon settlement, notably because of a possiblebankruptcy. The longer the maturity of the futures contract, themore serious the risk. To address this speci®c problem, exchangesthroughout the world have set up clearinghouses and marginmechanisms designed to force market participants to cover theirlosses as frequently as possible, in practice every day. The consequence of that intermediation has been the standardization of thecontracts to simplify procedures. It is naturally always possible toenter any type of contract off the ¯oor of an exchangeÐthat is, overthe counter (OTC)Ðin which case, each party faces a full-¯edgedcredit risk.The case of commodity futures is particularly instructive because it iseasy to understand why a corn grower would want to sell his or hersoon-to-come harvest now for delivery later. He or she gets the bene®tof a good price without the constraint of delivering corn that he or shedoes not own today but will harvest in the near future.However, in comparison to ®nancial futures, commodity futures ingeneral represent nowadays only a relatively minor fraction of the totalamount of futures traded throughout the world. We need to considerthese futures in more detail, and we will focus on stock index futuresthat have probably the largest applications in equity arbitrage:. A stock index future is a ®nancial futures; therefore, it is cash settled.
Its underlying is not one security in particular but a stock index.
. The contract is standardized in every respect. The underlying is
clearly identi®ed and unique. Expiration dates are ®xed accordingto a rolling cycleÐfor example, the third Friday of March, June,September, and December for the S&P 500 index futures.8
7
Note that a futures contract can also hedge a position that is not necessarily its underlying. In that case the correlation between that position and the futures underlyingalso plays a role in the overall pro®tability on the hedge.8If this Friday is not a business day, expiry is usually the day before. Index futures inGermany, Japan, and the United Kingdom also have a quarterly schedule. France hasa monthly schedule.There are usually ®ve expirations opened for trading at the same time. For example,if the ®rst expiry is March 2000, the last future available for trading is March 2001. The®rst expiry, called the front month, is by far the most liquid. The second expiry, the backmonth, starts trading actively around two weeks prior to the expiration of the currentfront month.
116
THEORY AND TOOLS
. An index futures contract is also characterized by its point value,
which is a coef®cient that indicates the variation in hard currencyassociated with a variation in the quoted price. For example, thepoint value is $250 for the S&P 500, therefore every variation of $1in the price of the futures contract induces a variation of $250 in thedollar value of the position.. The SQ price of an index futures contract is usually de®ned withreference to a particular time. For example, on the S&P 500, the SQis the value of the index computed with all the opening prices of itsconstituents on expiration date.9 The Nikkei 225 index expires on asimilar reference. Some European indices, however, have differentconventions. For example, in France and the United Kingdom, theSQ is computed as an average on the underlying index prices overa short period of time.. To ensure the ®nancial integrity of all participants, clearinghouseshave been instituted as necessary intermediaries, as we mentionedearlier. Every trade is actually carried out with a clearinghouse as acounterpart, which in turn has the legal responsibility and power tocollect dues. In order to do so, clearinghouses have instituted anautomatic margining system in which losses have to be systematically covered; if they are not covered, the positions are liquidated.10 Futures being a zero-sum game, payments to coverlosses on one account are necessarily credited to another accountwith an opposite position.. The exposure created by a futures position is highly leveragedbecause of the speci®cs of the margining system. Whereas buying
9
Note that this price has nothing to do with the value published by major data providers as the opening price for the index. The SQ price is known only when all stockshave opened, which may take several minutes. The opening price contributed byvendors is usually computed with opening prices for those stocks that have openedand previous closing prices for those that have not traded yet.10Naturally there are other measures to limit the amount of losses a participant issusceptible to accumulate. For example, open positions are monitored carefully andrestricted within certain limits.The failure of a member is a rare event but not unheard of. In 1995, for example, theBritish merchant bank Barings collapsed due to fraudulent trading by one of itsemployees on the Singapore Monetary Exchange (SIMEX). What happens in such acase depends on the exchange. Japanese exchanges, for example, collect a small fee onevery transactionÐtypically 0.001 percentÐsegregated in a dedicated safety fund tobe used to cover extraordinary losses. The Chicago Mercantile Exchange, on the otherhand, does not have a similar measure. Its rules state that members are responsiblefor every dollar of extraordinary losses in proration to their open positions.
CHAPTER 3 FUTURES AND RELATED INSTRUMENTS
117
$100 of stocks requires $100 of cash,11 buying the equivalent of $100of futures requires only a safety deposit that is much smaller thanthe traded nominal. If this deposit is, for example 20 percent, thesame gain of $1 will appear as a poor 1 percent return in the case ofa fully ®nanced stock position, and an honest 5 percent in the caseof leveraged futures. Therefore, futures are clearly attractive vehicles for fast, aggressive, and speculative gains.12. Like stocks, futures are also subject to speci®c rules and are settledwithin a speci®c settlement cycle. The actual ¯ows are due a certainnumber of days after the position has been traded in the market:D  1 in the United Kingdom and Germany, D  2 in the UnitedStates, D  3 in Japan, and so on.We will revisit many of these issues in detail; meanwhile Table 3-4gives a summarized view of the largest markets.
CLEARINGHOUSES, MARGIN CALLS, ANDDEPOSITSThe clearinghouse acts as an intermediary in each and every transaction and is legally entitled to take action to ensure the ®nancial integrity of the marketplace. Market participants are required to maintain acertain amount of ®nancial resources with their broker, and the brokerin turn is required to redirect a sizable fraction of these resources to theclearinghouse.There are two important notions in de®ning the de®nitions of theserequirements:. An initial requirement, always due at the opening of the position,
which represents the minimum balance to be maintained on theaccount
11
Or $50 in a margin purchase in the United States where the requirement is 50 percent.Essentially stock margin transactions are similar to futures in that they also raise theneed of a safety net in the form of regular margin calls. They differ, however, in thelevel of initial margin needed: Futures have a much lower requirement, allowinghigher leverage.12Note that we said ``the equivalent of $100 of futures.'' When buying $100 in stocks,the nominal settled is $100 for the delivery of the shares. When buying a futurescontract, the nominal of the transaction is the market price of the futures contracttimes its point value. This nominal, however, is not exchanged; only the depositrequirement is actually paid. Therefore, the $100 nominal in that case does not correspond to a physical ¯ow. It is sometimes referred to as a notional amount.
118
THEORY AND TOOLS
TABLE 3-4Index
Exchangea
PointValue
FTSE 100
LIFFE
£10
Quarterly
DAX 30
Eurex
25
Quarterly
Nikkei 225
OSE
¥1,000
Quarterly
Nikkei 225
SIMEX
¥500
Quarterly
S&P 500
CME
$250
Quarterly
CAC 40
MATIF
10
Monthly
abcdefg
Cycleb
ExpiryThirdFridayThirdFridaySecondFridaySecondFridayThirdFridayLasttradingday
SettlementRule
Tickc
D+1
£0.5
D+1
0.5
D+3
¥10
D+2
¥5
D+2
$0:5
D+1
0.5
SpecialQuotationEDSPaverageeIntradayauctionOpeningauctionEqual toOSE'sOpeningauctionAverage
MarginMarginRequirementd Calculation£3,000
SPANf
8,500
Otherg
¥1,000,000
Other
¥562,500
SPAN
$23,438
SPAN
2,000
SPAN
LIFFE, London International Financial Futures Exchange; Eurex, European Exchange; OSE, Osaka Securities Exchange; Simex,Singapore International Monetary Exchange; MATIF, MarcheÂ aÁ Terme International de France.Quarterly: March, June, September, December.The tick value is the minimum variation in the quoted price of the future.Initial marginÐthat is, the minimum amount of money required on the account to authorize trading for one contract.EDSP, Exchange delivery settlement price.SPAN, Standard Portfolio Analysis of Risk, a margining system designed by the CME and very broadly used by clearing corporationsworldwide.The Eurex runs its proprietary risk procedure, called ``Risk-Based Margining.'' Margin requirements are recomputed every day for severalclasses of assets, in a way similar to SPAN evaluations.
(Source: Exchanges respective Web sites, www.liffe.com, www.eurexchange.com, www.ose.or.jp, www.simex.com, www.cme.com,www.matif.fr, and Bloomberg.)
. An incremental requirement, payable or receivable depending on
market movements
These two quantities carry several names, like initial and maintenancemargin, or initial deposit and minimum margin call. Beyond the labels, thedifference resides in the fact that the initial margin covers the positionas it is opened and must always be met after subsequent marketmoves. The maintenance margin, by contrast, is the minimum incremental amount that can move in and out of the account in order tokeep the account in good standing. For example, the requirements forthe CME S&P 500 are as follows: $23,438 per contract upon initiation,$18,750 for maintenance for outright positions.13 The distinction of outright position is necessary because speci®c rules allow investors todecrease their margin calls if a position results in lower market riskthan a straightforward futures exposure. For example, a calendarspreadÐthat is, long one expiry and short anotherÐis considered13
An outright position has one leg only, in contrast to a spread that has two oppositelegs.
CHAPTER 3 FUTURES AND RELATED INSTRUMENTS
119
much less risky and the deposit requirement is smaller: $219 uponinitiation and $175 for maintenance.The existence of margin calls is an important parameter to considerwhen trading futures. The cash ¯ows necessary to maintain a futuresaccount generate treasury movements that need to be accounted forcarefully, if only because of the ®nancing entries they create.14 It istherefore necessary here to achieve a clear understanding of the margining system. By and large all margining systems function in a similarway, so we will consider from now on the example of the S&P 500futures to illustrate our discussion.Consider an investor trading futures with the following transactions:. On January 14, she has no preexisting position and she buys 100
S&P 500 futures at $1,250, with the market closing at $1,255.next day with the market up, she decides to liquidate 25futures at $1,265, but at the and of the day, buys 50 more on theclose at $1,240.. Finally, on January 16 she liquidates everythingÐthat is, 125 contracts, at a market price of $1,230.. The
Table 3-5 summarizes his or her trading activity. Every day this investor faces potential pro®ts or losses due to the difference between themarket closing price and the historical trades in her portfolio. In addition, because she is still trading during the period, some of these pro®tsor losses actually materialize.The process of comparing an existing position to its value in themarket is known as a mark-to-market valuation. This is the most basicfunction performed by a clearinghouse, after which it is in a position tocall some accounts on margin while crediting others by an equal amount.
TABLE 3-5Date
Position(at Open)
Buys
Sells
Close
Jan. 14Jan. 15Jan. 16
Ð100125
100 @ $1,25050 @ $1,240Ð
Ð25 @ $1,265125 @ $1,230
$1,255$1,240Ð
14
Remember that from our trading perspective, every dollar needed is borrowed; therefore, margin calls are borrowed or invested at market rates.
120
THEORY AND TOOLS
TABLE 3-6
Pro®t/Loss
Margin Calla
AccountBalance afterMargin Calls
$2,343,800
$125,000(1)
$2,218,800(2)
$2,343,800
$2,929,750
$218,750(3)
$804,700(4)
$2,929,750
Ð
$312,500(5)
$2,617,250(6)
Ð
Beginning-ofDay Position
End-of-DayPosition
InitialRequirement
Jan. 14
Ð
100
Jan. 15
100
125
Jan. 16
125
Ð
Date
a
The convention adopted here is the investor's view: a negative margin call is cash going out the door.
Table 3-6 shows the margin calls and the account balance. The®gures result from the following calculations:. (1) At the end of the day, the investor has a potential pro®t of
$125,000 because she bought at $1,250 a position valued at $1,255($5  100$250 $125;000, $250 is the S&P 500 point value). Hermark-to-market valuation is positive. She has not locked this pro®t,however, because she still holds the position: This is an unrealizedpro®t.. (2) $2,218,800  100  $23,438$125,000  initial requirement foropening the position, minus the unrealized gain. This amount islarger than the maintenance requirement, $18,750, so it is claimedin full by the broker.. (3) The in and out on 25 futures during the day at $1,240 and$1,265 generated 25  $25  $250  $156,250. The unrealized pro®ton the 25 lots bought at the end of the day is 0 because they weretraded at $1,240. Finally, the 100 futures carried overnight have amark-to-market of $375,000  $15  100  $250). In total, theloss for the day is $156,250 $375,000  $218,750.. (4) The total initial requirement for 125 futures at the end of theday is $2,929,750. Considering the investor has a total mark-tomarket of $218,750, she has to pay $2,929,750( $218,750)$2,343,800  $804,700.. (5) The investor is closing 125 futures at $1,230. These futures weremarked to market at $1,240 the previous night. The realized loss istherefore: $10  125  $250  $312,500. (6) The investor receives what is remaining on the account as a®nal ¯ow, that is the previous balance $2,929,750 minus the currentloss $312,500.. The total loss for the whole period is given by the sum of all margincalls, or by the sum of all daily losses, in both cases $406,250 here.
CHAPTER 3 FUTURES AND RELATED INSTRUMENTS
121
Note that all the margin calls paid out during the period come backexcept for the realized loss.15. The total interest charge for the period is $728, which is relativelyminor.16 However, suppose that January 16 is a Friday and the®nal trading day is therefore postponed until January 19. The®nancing charge then becomes $1,382; it is still low but it isgrowing fast.We have assumed here that payments are all settled in cash, butthis is not the general case. In fact, because of the interest charges onmargin calls, cash is very rarely used, and most clearinghouses acceptalternative assets such as bonds issued by their domestic government.Most of them also accept U.S. treasury bonds. Some also acceptdomestic shares or other forms of collateralÐletters of credit, forexample.In any case, there is a direct cost associated with a deposit requirement, even when cash is not used. If an arbitrageur wants to depositbonds or stocks, he or she has to borrow them.17 Against this borrowing, he or she will have to deliver a cash collateral, but the total cost ofthe operation is much lower than depositing cash directly at the clearinghouse. If he or she wants to use a letter of credit, the issuer of theletter will also claim a fee for this service. If the arbitrageur held somestocks, which is often the case by nature of the position, he or she coulddeposit them and apparently save the fee. At the same time he or shewould not be able to lend these stocks, which would result in anopportunity cost.Because of the amounts involved, it is extremely important for anarbitrageur to secure the cheapest deposit possible and to make sure15
Note also that on a given day, the total pro®t and loss for the position is the sum of arealized part, arising from deals effected on that day, and an unrealized part, theoutstanding position carried over from the previous day.16The calculation is as follows: Assuming a 5 percent interest rate, $2,218,800 is borrowed from January 14 to January 16 when the position is closed. The interest chargeis ($2,218,800  5 percent  2 days) / 360 = $616. An additional $804,700 is borrowedfrom January 15 to 16, at a cost of $112. The total is $728. Each additional day theposition is kept costs $420 if the market is unchanged.The reason this is a charge is that clearinghouses, and more generally brokers, areusually not allowed to pay interest on the money they receive as collateral for derivative positions. This interdiction arises from the fact they are not ®nancial institutions in the business of managing money. Therefore, depositing money as a result of amargin call generates a pure ®nancing loss.17If he or she buys them, he or she carries the market risk over a period that can besigni®cant. Borrowing does not transfer market risk, which remains with the lender.
122
THEORY AND TOOLS
that the ¯ows generated by the market movements get accuratelydescribed and accounted for.
GENERAL MARGIN CALL FORMULAWe have explained how a margin call is computed on the basis of amark-to-market valuation, but we implicitly effected a LIFO matchingbetween open positions.18 In computing a pro®t and loss for the day onJanuary 15, we matched the 25 lots sold to the 50 lots bought on thesame day. That is, we matched the latest purchase deals to offset thesale of 25 lots. Alternatively, we could have chosen the long positionoutstanding from the previous day, and matched the sale against these.This method would have been a FIFO matching because the positionopened on the January 14 preexisted the purchase made on January 15.The situation is therefore more complex than it appears. Indeed,there are essentially two generic ways to manage a futures account:. Position carried at market, no historical matching: In that scenario, the
outstanding position is reevaluated every day at close. Deals for aparticular day are matched entirely, and they create a pro®t or aloss that is fully re¯ected in the margin call.19 This is essentiallywhat we did above. In that situation, the historical informationabout past transactions is irrelevant. All deals prior to today areevaluated as if they had been executed at yesterday's closing price.The margin call encompasses both realized and unrealized P&Lwith no further distinction.. Position carried at cost, speci®c matching rule: In that scenario, alldeals are carried at cost and ranked according to a speci®c key.This ranking is necessary to pair deals and offset sales and purchases in order to leave the account with one outstanding aggregate position, long or short.20 In a FIFO matching, the oldest deal isretrieved and matched ®rst. In a LIFO matching, the most recentdeal is retrieved and matched ®rst.21 Historical information for18
LIFO = Last in ®rst out, as opposed to FIFO, ®rst in ®rst out.Pro®t and loss will be abbreviated P&L from now on.20This procedure is called netting, and is not always compulsory or automatic. In Japan,for example, it is possible to carry both long and short positions on the same account,without their being matched.21For example, assume that we carry a long position, and today we sell one contract. AFIFO matching pairs today's sale with the oldest purchase available. A LIFO matching would pair the same sale with the most recent purchase.19
CHAPTER 3 FUTURES AND RELATED INSTRUMENTS
123
each deal is critical in order to effect the matching properly. Themargin call is the sum of realized and unrealized P&L, and it isnaturally identical to the previous method. However, the key difference is the amount of realized P&L, which may have a signi®cant impact on the taxation of the position.22Differences between the two methods are not fundamental if weneglect the problem of accounting and taxes because overall the monetary ¯ows are identical. They are only distinct formats presentingessentially the same information.The following equations present a formalization of these results:23XQ  P5PLt 1!t  PV4Pt  Ct Pt 1  Ct 1MCt 1!t 
today0 s deals
0Pt 1
ifj Pt 1
Pt   IM  PLt 1!t j < MM
Pt   IM  PLt 1!t
otherwise
unrealized MCt 1!tPLrealizedt 1!t  PLt 1!t
 total P&L from yesterday to today23 margin call from yesterday to today initial margin requirement24 maintenance margin requirement future point value outstanding position and closing pricetoday, respectively25Q; P quantities and prices for today's deals,respectivelyrealized,  realized and unrealized parts ofunrealizedPLt 1!t 26
where PLt 1!tMCt 1!tIMMMPVPt ; Ct
22
Another form of matching is called best: The deals are matched in order to realize themaximum amount of pro®t, or the minimum amount of loss. Again this mode has animpact on tax calculations.23PLt 1!t > 0 indicates that the position shows a pro®t. By the same token, MCt 1!t > 0indicates a positive ¯owÐthat is, a payment to the account holder.24In some rare cases, IM is a percentage of the current market price and is not a ®xedamount.25By convention P is an algebraic number: P > 0 indicates a long. Along the same lines,Q is also algebraic and follows a similar convention: Q > 0 indicates a purchase.PThese conventions imply that Pt Pt 1  today0 s deals Q.26
unrealizedNote that getting an exact expression of PLrealizedis not possible witht 1!t and PLt 1!tout a historical breakdown of deals.
124
THEORY AND TOOLS
The margin call is