therefore composed of two terms:. (Pt
Pt 1   IM: minimum required balance for the current size ofthe position. PLt 1!t : dependent upon the day-to-day trading activity and market levelAs an application, consider our previous example and the margincall from January 14 to January 15. Here t  January 15:. Outstanding position: Pt 1  100, Pt  125, IM  $23,438. Today's deals: QB  50, PB  $1,240, QS  25, PS  $1,265. Closing prices: Ct 1  $1,255, Ct  $1,240
Therefore:PLt 1!t  250  125  $1,240
100  $1,255
50$1,240  25  $1,265
which becomes:PLt 1!t  250  $155,000
$125,500
$62,000 + $31,625)  $218,750
And the margin call is given by:MCt 1!t  100
125$23,438
$218,750 =
$804,700
Both ®gures are equal to those we obtained before.Our presentation can be simpli®ed further because we adopt theperspective of an arbitrageur, and we do not expect to have any trading activity once the positionis set up. UnderPP this steady-state assumption, Pt  Pt 1 andtoday0 s sell Q  P today0 s buy Q  P  0 becausethere is no trading activity. The above equation becomes:27MCt 1!t  PLt 1!t  PV Ct
Ct 1 Pt
The margin call from yesterday to today is directly proportional to thevariation of the market.28 This result is fundamental because it givesa precise expression of the necessary ¯ows to maintain the futureposition, and shows that these ¯ows depend on market movements.27
For all practical purposes, the term MM is negligible in front of(Pt 1 Pt   IM  PLt 1!t in the calculation; therefore, we assume the condition(Pt 1 Pt   IM  PLt 1!t > MM is always true.28It is certainly a natural and intuitive result, but interestingly it holds despite the rathercomplex margining structure.
CHAPTER 3 FUTURES AND RELATED INSTRUMENTS
125
We will develop further this argument in the examination of indexarbitrage.
CUSTOMER MARGIN CALLS VERSUSEXCHANGE MARGIN CALLSSo far we have not made any distinction among market participantswith respect to clearing procedures. In practice only exchange membersare allowed to maintain an account directly with the clearinghouse.Other institutions have to have intermediaries to clear their trades.Therefore, the margin calls managed by the clearinghouse are infact aggregated over a large number of individual accounts, as shownin Figure 3-3. In that situation, the failure of a member has a potentiallymuch more devastating effect than the failure of a single participant,which explains the high level of regulation imposed on ®nancial institutions engaged in trading activities for customers.29At the same time, to provide members with a safety cushion, therequirements on customer margin calls are often more stringent thanthose set by the exchange for its members.30 Typically customers'deposits are higher and must be more liquid. The extreme example isan individual investor, for whom the types of deposits allowed to tradefutures are often cash or money market funds only.
Exchange
Member A
Client a
Client b
Client cFigure 3-329
Member B
Client a
Client b
Client cClient versus Broker MC
Notably the diverse regulations call for a complete segregation of customers' assets toavoid a situation in which a rogue institution settles and disguises its own losses atthe expense of its account holders.30In some examples, Japan notably, an additional protection feature is that customermargins are due one day before exchange margins, and paid one day after.
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THEORY AND TOOLS
STANDARD PORTFOLIO ANALYSIS OF RISKThe Standard Portfolio Analysis of Risk (SPAN) is a risk evaluationsystem originally developed by the Chicago Mercantile Exchange in1988. It was designed to solve simple problems related to marginingrules for derivative portfolios.Consider, for example, a portfolio with a long position in a calloption, and a short in a futures contract on the same underlying. Longpositions in options do not create any margin requirement becauseonce the option is paid for, the holder has no subsequent risk. Onthe other hand, the short leg of the position presents, in itself, a signi®cant exposure and should consequently be margined appropriately.However, when put together, it can be argued that the call ``protects''the short to some extent, and therefore the whole portfolio is less riskythan an outright futures position. In fact, if the call is in-the-money, theportfolio is indeed completely protected and presents a very low levelof risk. Regular margining rules would charge this portfolio with arequirement derived from the futures position although it may bealmost perfectly hedged in practice.Therefore, applying margining calculations instrument by instrument lead to requirements much too high and inappropriate withrespect to the actual level of risk. An alternative approach is to considereach portfolio as a whole and run risk scenarios to identify potentiallosses arising from different market situations. This is exactly whatSPAN does, running the following 16 scenarios31:. Futures unchanged, volatility up and down
. Futures up 1/3 range, volatility up and down
. Futures down 1/3 range, volatility up and down. Futures up 2/3 range, volatility up and down
. Futures down 2/3 range, volatility up and down. Futures up 3/3 range, volatility up and down
. Futures down 3/3 range, volatility up and down. Futures up extreme move
. Futures down extreme move
The margin requirement is then set to the maximum loss under any ofthe 16 scenarios.
31
The futures ``range'' is the amplitude of the price movement creating a change in thevalue of the position equal to the maintenance margin, and the ``extreme move'' istwice that range. Volatility analysis is meant to capture the risk on option positions.
CHAPTER 3 FUTURES AND RELATED INSTRUMENTS
127
From a local initiative, SPAN has expanded to become an industrystandard, and nowadays numerous derivative exchanges use it to evaluate margin requirements. SPAN has been adopted both by clearinghouses for their members, and by brokers-dealers for their customers.Under a SPAN margining system, our simpli®ed margin formalismdoes not necessarily apply because of cross hedging between differentinstruments. It is valid as a ®rst approximation if futures constitute themain source of derivative risk in the portfolio.
SPECIAL QUOTATION CALCULATION ANDCONVERGENCE REVISITEDIn the case of index futures, there are two primary types of SQ32:straight reference and average. A straight reference is obtained bycomputing the index with a particular type of price for each stock. Inthat category, we ®nd the S&P 500, Nikkei 225, and DAX 30. The ®rsttwo have an SQ equal to the price of the index computed using the openingprice of each component on expiration day, and the DAX futures settles onan intraday auction. An average SQ is obtained by taking the averageof index prices measured periodically over a short period of time.Consider, for example, the of®cial de®nition as given by the Londonfutures exchange for FTSE 100:Exchange Delivery Settlement Price (EDSP): The EDSP is based onthe average values of the FTSE 100 Index every 15 seconds inclusively between 10.10 and 10.30 (London time) on the Last TradingDay. Of the 81 measured values, the highest 12 and lowest 12 will bediscarded and the remaining 57 will be used to calculate the EDSP.Where necessary, the calculation will be rounded to the nearest halfindex point.33France falls into that category as well with an average on 20 minutes atthe end of the expiration day.To understand why these de®nitions are what they are, we needto realize that an SQ must be a ``realizable'' price because futuresholders must be able to transfer their positions to cash on expiration.34
32
As a reminder, the special quotation is the ®nal settlement value of a futures contract.LIFFE Web site.34Realizable means that the price can be reproduced through a reasonably low-risktrading strategy. This is the practical consequence of the absolute convergence thatexists between a futures contract and its underlying.33
128
THEORY AND TOOLS
Therefore, the ®nal reference must be chosen so that all outstandingpositions can be transferred to the underlying stocks with anadequate level of liquidity, without any dramatic impact on themarket.The closing price, which looks like a natural choice, is not a validoption because if an important unbalance materializes betweenbuyers and sellers, the trading system may be unable to matchthem properly. In that case, the ``fair'' price of the stockÐthat is,the price that matches all present interestsÐis impossible to establish,and the SQ is not representative of the latest market conditions. Inaddition, investors are potentially left with sizable unexecuted ordersthat need to be carried over to the next day. Along the same lines,expiry on a single point in time during the day is dif®cult to implement with continuous trading systems, except if the system is appropriately equipped to handle exceptional punctual levels of liquidity,as, for example, in Germany.Overall the solution that has been adopted is to spread liquidityover a short period of time. In the case of an SQ at open, the aggregation of orders is such that liquidity can ¯ow to and from a particularstock before it actually opens. Because it is an auction, all orders arenecessarily executed removing any execution risk.35 The other implementation calls for an average over time so that the market can absorbthe exceptional liquidity despite the fact that the process takes placeduring a regular continuous session without any auction. In otherwords, an SQ at open spreads liquidity over price, whereas an averagespreads it over time.Each of the two methods provides continuity and liquidity for SQcalculation and realization, and there is not much of a choice beyondthem. It is interesting to note the repartition of the two groups: theUnited States, Japan, and Germany have opted for an auction reference while the United Kingdom and France have chosen an average.It is certain that transparency and simplicity are two powerfulrequirements for investors, and in these respects, opening SQs mayappear less representative than averages but are far simpler and saferto manage.What are the consequences of these de®nitions for investors andarbitrageurs? In the case of an opening, SQ realization is straightfor35
It is extremely rare but not impossible for a stock not to open at all because of amarket imbalance. We will see an example of that later in the chapter dedicatedto index arbitrage. Therefore, the nonexecution risk still exists, but it is verysmall.
CHAPTER 3 FUTURES AND RELATED INSTRUMENTS
129
ward, whereas for an average things get much more complex. In orderto realize the SQ price, a trader needs to replicate every price in thecalculation sequence. For example, in the case of the FTSE 100, thereneeds to be partial executions every 15 seconds. The whole positionmust be partitioned into 81 parts, each executed separately, with thehope that the truncature of the 12 highest and lowest will not result in alarge deviation.36 Obviously the overall risk is signi®cant, and managing an expiry on an average requires a lot of skill and a fairly decentnominal.37It would already be dif®cult to calibrate an execution on 81 printsin a friendly market, but it gets even more risky because futurestrading is a zero-sum game: For each buyer, there is a seller ofequal size. Therefore, each player buying cash on expiration tocover for a long futures expiring is potentially confronted by a sellercovering a short position. This remark leads to the following conclusion: If a trader is trying to manage an expiry by replicating thecalculation sequence one way, somebody else is trying the otherway.38 It is exactly like two people pushing the same sliding wallfrom each side at the same time periodically. The strongest eventuallywins; however, the weakest may push earlierÐor laterÐfrom time totime and signi®cantly ``disturb'' the mechanism the other is trying todevelop. Thus, if trading itself is strictly zero-sum, it is not true forexpirations: On average both lose because they are not the only onesanyway at a given time.39 For all these reasons, it is always wise toconsider all available options before deciding to ``shoot'' for cash onexpiration on an averaged SQ.
36
The truncature problem is subtle. Skipping one print on the expectation that it will beeliminated is a risky proposition at the beginning of the interval, much less so at theend. Experience is a powerful help in these cases.37A large nominal to execute allows much more ¯exibility during the 20-minute interval because slices can be of different sizes and timed precisely. With the execution of asmall nominal, it may not even be possible to slice executions into 81 parts, and only apassive strategy is possible.38The good thing is that if one arbitrageurÐtypically an index arbitrageurÐdecides tointervene massively in the cash market at expiry, others involved in similar strategiesmost probably have the same interest for the same reasons. It is fair then to expect asigni®cant aggregation of trading power in the same direction. However, arbitrageursare not the largest participants by far, so it is presumptuous to make hasty conclusions even in that case.39This statement is naturally an exaggeration, but it is extremely dif®cult to manage anaverage expiration ``properly.'' If there are indeed several players in opposite campsand fairly large equal trading power, then most probably everybody loses.
130
THEORY AND TOOLS
FINAL SETTLEMENTSThe margin call is known at the end of each trading day, once theclosing price of the day is of®cially available from the exchange. Onexpiration day, the position is liquidated with the ®nal reference pricebeing the SQ price. If we go back to our previous example, we are long125 futures on January 16. We assume that the closing price on January16 is $1,235 and that we hold the 125 futures instead of selling them.On January 17, the futures contract expires at $1,241.54, and we let theposition expire, as shown in Table 3-7.The margin call (1) is derived from the formula we introducedpreviously: The closing price changes from $1,240 to $1,235, and 125futures lose $5  $250  125  $156,250. The account balance (2) is identical to what it was on January 15: the additional amount injected bythe margin call into the account has been consumed by mark-to-marketlosses.The ®nal margin call is the sum of the mark-to-market P&L ofthe position and of its total unused deposit. From $1,235 to$1,241.54, the position made $204,375, which represents the P&Lpart. The sum $2,929,750 represents the excess deposit because theposition has been liquidated. Therefore, the total margin call is$3,134,125, which is ¯ow (3). The general mechanism of ®nal settlement is exactly equivalent to a ®nal closing saleÐor purchaseÐofthe position.
EXPOSURE VERSUS INVESTMENTFutures, and more generally derivative positions, have an interestingcharacteristic: Because they are traded on margin, the amount ofmoney needed to enter a position is much smaller than the nominalrisk created by this position. For a stock transaction, every variationin the price of the stock results in a 1-to-1 variation in the value ofthe position. If the stock is paid in full, exposure and investment areTABLE 3-7
Date
PositionBeginning ofPositionInitialDayEnd of Day Requirement Closing Price Margin Call
AccountBalance
$156; 250 1
$2,929,750 2
$3,134,125 3
Ð
Jan. 16
125
125
$2; 929,750
Jan. 17
125
Ð
Ð
$1; 235$1; 241:54
CHAPTER 3 FUTURES AND RELATED INSTRUMENTS
131
equal and move together. In the case of a margin trade, we have tosettle only 50 percent of the full amount to enter the same position;the remaining 50 percent are borrowed from the institution managing the account. In that case, our pro®t is still exactly equal to anyvariation in the price, but our investment is half what it was.For futures this difference is even higher because margin requirements are much smaller than 50 percent. In the case of the S&P 500, arequirement of $23,438 is less than 7 percent of the nominal of a futurescontract at $1,400: an investment of $23,438 allows an exposure of$350,000. This well-known property is called ®nancial leverage, whichwe de®ne here as the ratio between exposure, that is, the nominal atrisk, and investment, that is, the capital required. Table 3-8 is the current leverage of different futures.Leverage is particularly important for many investors and particularly speculators. However, from an arbitrageur's point of view it is agiven of no particular importance beyond the underlying marginingsystem:. Costs associated to the carry of the position are generated by mar-
gin requirements only; therefore, what is really important is theabsolute amount of collateral requested, not the leverage resultingfrom it.. The ultimate object of our attention is exposure. In an index arbitrage position, for example, we need to compare what is comparableÐthat is, exposure on each leg, not invested capital.. Arbitrage is not speculation but a trading strategy, usually practiced in a speci®c context where capital is readily available. Asa device to reduce capital requirements, leverage is thereforerelatively unimportant.
TABLE 3-8Current MarketPrice
FTSE 100DAX 30
Leverage Ratioa
Point Value
MarginRequirement
ContractNominal
£6,100
£10
£3,000
£60,000
20:1
7,500
25
8,500
187,500
22:1
Nikkei 225 (OSE)
¥20,000
¥1,000
¥1,000,000
¥20,000,000
20:1
Nikkei 225 (SIM)
¥20,000
¥500
¥562,500
¥10,000,000
18:1
S&P 500
$1,350
$250
$23,438
$337,500
14:1
CAC 40
6,000
10
2,000
60,000
30:1
Index
a
Note that the leverage ratio depends on the current market price because margin requirements are usually constantthrough a long period of time.
132
THEORY AND TOOLS
CROSS LISTINGSWhat is true for stocks is also true of futures: More and more exchangesworldwide cross list identical or similar contracts. There are two criticaldifferences, however, with respect to cross-listed derivatives comparedto stocks.First, cross-listing initiatives for derivatives usually do not needany sponsorship because of the nature of the instruments: No corporation is in a position to claim ``ownership'' of a futures contractand sponsor a cross listing. Furthermore, many of these listings effectively result in additional competition for liquidity, which may ormay not be well perceived by the institution that listed the originalfutures.Second, the clearing part is critically important with respect toderivatives because of margin requirements. Suppose that the samecontract is listed in the United States and United Kingdom, or in theUnited Kingdom and continental Europe. If the contracts are recognized as fully fungible, an investor can intervene indifferently onboth markets with similar requirements, and all trades can be nettedon a single account regardless of where they were actually effected.On the other hand, if the contracts have similar characteristics butare not fungible, each has to be margined and managed independently.In characterizing two fungible contracts, there are four importantconditions:. Identical underlying, which seems a very natural prerequisite.
. Identical SQ de®nition, for its expiry dates as well as calculation
method.
. Similar characteristics, not directly related to underlying, expiry, and
SQ prices: point value, deposit requirements, settlement cycle, currency used for margin calls, and so on.. Reciprocal clearing agreements between exchanges. This is actuallythe most important condition to ensure that the contracts can betraded indifferently. There may be cost differentials associatedwith trading one as opposed to the other, notably in terms ofbrokerage fees, but they are automatically netted on clearingstatements.The last condition is the most dif®cult to satisfy because it requiresmore than a paper agreement: There must be an electronic link
CHAPTER 3 FUTURES AND RELATED INSTRUMENTS
133
between the different clearinghouses. This is exactly why fungibility isfar from being the rule, and remains still an exception.40 Most crosslisted contracts are listed on different exchanges without being fungible. In these cases they actually compete for liquidity because investorshave an objective choice between two distinct instruments, usuallydesigned to be very close.There can be two main reasons for cross listing, quite different fromthose underlying corporate decisions to list a stock in different countries:. Market accessibility: For structural reasons, a market is dif®cult or
expensive to access. Another exchange with some type of competitive advantage, usually in the same geographical area, willseize the opportunity and list a contract under conditions thattend to improve accessibility. It is easy to ®nd examples of thiswith Japan that long remained an inef®cient fortress. TheSingapore Monetary Exchange realized there was an opportunityand listed a contract on the Nikkei 225 index, which in effectbecame a direct competitor to the ``of®cial'' contract listed inOsaka.41. Round-the-clock trading: The need for continuous trading has beenclearer and clearer over the years, notably because of unavoidable time differences between geographical zones. Even though itis unrealistic to expect liquidity to be equal in all parts of theworld, at least the opportunity to trade, even small-size orders,is better than no market at all. The Chicago Mercantile Exchange,for example, listed a Nikkei 225 contract to address this concernand made it a dollar-denominated contract to solve at the sametime the problem of currency conversion. Other answers to thesame question have prompted the development of electronic
40
In fact, no index futures contract in the list of those we consider has a fungiblecross-listed contract. This situation is more frequent, however, on interest ratefutures. The euroyen contracts listed on the Tokyo International FinancialFutures Exchange (TIFFE) and the London International Financial FuturesExchange (LIFFE) are fully fungible, as are the eurodollar on the ChicagoMercantile Exchange (CME) and Singapore International Monetary Exchange(SIMEX).41For several years the Singapore futures expired on a reference different from thatof its Osaka counterpart, probably to avoid being so obvious a competitor.Eventually the SIMEX revised its rule and adopted the same reference. Still thetwo contracts are not fungible, notably because of their respective point values.
134
THEORY AND TOOLS
trading systems like Globex offering a wide range of contractsduring and after regular hours.42Let's explore now two particular examples of cross-listed indexfutures.
Nikkei 225 on the Osaka Stock Exchange versusNikkei 225 on the Singapore InternationalMonetary Exchange
Japan has traditionally been very rigid about and very protective of itssecurities industry. Policymakers today face a considerable task inreforming and deregulating, starting with the gigantic and almightyMinistry of Finance. Transactions on stocks and futures, for example,were taxed signi®cantly, and brokerage commissions for trading stocksor futures were set for the whole industry and largely nonnegotiable.43Along the same lines many other parametersÐtypically marginrequirementsÐhave long been relatively high compared to other ®nancial centers.44The combination of high commissions and high holding costsmade Japan an expensive market. This was seen as an opportunityby Singapore, which listed an identical Nikkei 225 contract in a ``friendlier'' environment:. The contract has the same maturity cycle and SQ calculation. Its
actual de®nition states equality between its SQ and OSEs. It isconsequently also settled in Japanese yen.. Deposit requirements do not call for cash. SIMEX allows othersecuritiesÐU.S. treasury bonds, for exampleÐand other types of42
The S&P 500 future, for example, is available on Globex but only after normaltrading hours in order to avoid direct competition with the ¯oor. It opens at4:45 p.m. EST, 30 minutes after the ¯oor closing, and closes in the morning at9:15 a.m. EST. Globex also accepts orders for the E-Mini, an S&P 500 contractwith a point value of $50. E-Mini is traded only on Globex and order size iselectronically limited to 30 lots to deter trading by professionals and institutionalinvestors.43The same legal situation prevailed in the United States at the beginning of the 1970s.There have been considerable changes in Japan over the last decade, so manyÐbutnot allÐof these direct or indirect costs have disappeared today.44In particular, until a few years ago the OSE requested its members to deposit cash fora sizable portion of the deposit requirements. In turn, brokers also had to request cashfrom their customers for an even higher fraction.
CHAPTER 3 FUTURES AND RELATED INSTRUMENTS
135
collateralÐletters of credit. Holding a SIMEX position is thereforesigni®cantly cheaper than holding an OSE position.45. The size of the contract is smaller (point value of ¥500 instead of¥1,000), and its tick value is smaller as well (¥5 instead of ¥10).Literally: 1 OSE  2 SIMEX.. Trading limitsÐhighest and lowest prices allowed dailyÐarebeyond those set by the OSE rules.46. A few years ago SIMEX decided to open its Nikkei 225 ®ve minutesbefore the OSEÐ8:55 a.m. instead of 9:00 a.m. In addition, for along time SIMEX has been closing 15 minutes after the OSEÐ3:15 p.m. instead of 3:00 p.m.47. Commissions in Singapore are not regulated at all as opposed tothose in Japan.. In order to attract more than futures traders, SIMEX also listed awhole series of options on the Nikkei 225, identical to the OSE'swith minor differences, notably lot sizes.The natural question at this point is whether the SIMEX initiativedrained signi®cant volume away from Japan, and as of today theanswer is most de®nitively yes. As an example, Figure 3-4 shows therepartition of trading volume between the two marketplaces through1999.48 Even on a longer time frame, and considering open interestinstead of traded volume, SIMEX seems to have unequivocally conquered its place under the sun, as shown in Figure 3-5.Overall market shares look reasonably stable, with SIMEX holdinga fair 25 percent of the total volume and open interest. It can be arguedthat 25 percent is indeed a sizable presence in what could have beenvirtually a captive market, and one can wonder to what extent the OSEsimply failed to adapt and contain this slow drain. Nonetheless, bothmarkets are now clearly established in the marketplace and trade with
45
This induced a difference between the trading prices of the two contracts. They werecontinuously trading at a ¥5 difference. This discrepancy somewhat disappearedwhen the OSE lowered its deposit requirements.46This particular point was made very clear in 1995 after the Kobe earthquake andwhen the Barings collapsed. The SIMEX contract traded at its limit down price severaltimes during that period when the OSE futures were also limit down at a muchhigher price. The limit on SIMEX was ¥1,000 below the previous day's close, whereasthe OSE's was ¥600.47More recently the OSE extended trading hours to 3:10 p.m., prompting SIMEX toextend further from 3:15 p.m. to 3:25 p.m. in a move to offer the extra mile of trading.48The trading volume of SIMEX has been divided by 2 in the chart to re¯ect its nominalsize compared to the OSE.
136THEORY AND TOOLS
Figure 3-4 Volume on the SIM and OSE(Source: Compiled by author from Datastream data.)
SIM verses OSE Open Interest(5-day moving average)
CHAPTER 3 FUTURES AND RELATED INSTRUMENTS137
Figure 3-5 Open Interest on the SIM and the OSE(Source: Compiled by author from Datastream data.)
138
THEORY AND TOOLS
a fairly adequate liquidity. Figure 3-6 shows the average price difference between the two futures over 1999. Despite the 10 minutes difference in closing timesÐOSE closes at 3:10 p.m. and SIMEX at3:25 p.m.Ðprices are fairly close. The average over 1999 was ¥3.4,showing that OSE is trading at a small discount compared to theSIMEX.49From an arbitrageur's perspective this situation is extremely convenient and comfortable. Both markets are essentially open over thesame periods, and one is a ¯oor (SIMEX) while the other is electronic(OSE). A trader gets all the possible bene®ts: a dual source of liquidity,the user-friendliness of an electronic system on the main market, andthe volatility and opportunities of a ¯oor.
Nikkei 225 on the Osaka Stock Exchange andNikkei 225 on the Chicago Mercantile Exchange
The CME contract on the Nikkei 225 is an interesting example ofunconventional contract. It was primarily listed for conveniencebecause of the time difference between the United States and Japan.Naturally, because it attracts investors mostly outside Japan, its liquidity is very poor, but again it offers an option for late trading, which isbetter than no market at all.50It is unconventional because it is an ADR-like futures contract:Listed in U.S. dollars with a point value of $5, all ¯ows are settled inU.S. dollars, but the contract is quoted in Japanese yen. This last pointis particularly important