theQuickvariety of assetspresent and the sizeof the correspondingvariance/covariancematrixHistorical price/rateHistorical P&L ofdata of key marketportfoliovariables or J.P.Morgan's risk datasetsto the extent that theycover all relevantassets in the portfolio
Historical price/ratedata of keymarket variables
All historical marketdata that arenecessary toestimate theparameters of thestochasticprocesses used inthe risk modelingPrecision of riskDepends on theGood if the portfolio Good if theBetter precision to themeasurevalidity of its maincomposition ishistorical path isextent that theassumptions ± stability relatively constantrepresentative ofstochasticof asset variances and over time and there all future marketsimulation is morecorrelations, normality is little ageing effect behaviorrealistic andof return distributions,captures morelack of optionmarket interactionscomponentsand portfolio detailsBest choice when . . . Portfolio assets/Portfolio mix doesPortfolio hasPortfolio hasliabilities have nonot change overoptionality andsubstantialoptionality eg atime and ageinghistorical path isoptionality and thereportfolio of stocks,effect is minimalrepresentative of all are many possibleFX and/or spotpossibilitiessample paths, eachcommoditiesrepresenting adifferent riskWorst choice when. . . Portfolio hasPortfolio mixHistorical path isThe portfolio issubstantialhas changedan outlier of allenormous and theoptionalitysubstantially overpotential outcomespotential sampletimepaths are many;hardwareconstraintsRepresentative user J.P. Morgan,Merrill Lynch uses a Some commercialSalomon, Bearquantitative portfoliocombination of therisk managementStearns, Enronmanagershistorical methodsystems vendors(actual P&Lsuch asdistributions) andAlgorithmics, LOR/stress testing for risk GB and Sail®sh offermeasurement. Seehistoricalthe company's 1995 simulations as oneannual report forof the optionsdetailsNote: ``Cash¯ow mapping'' indicates a simplifying treatment of cash ¯ows within a portfolio, assumed to settle on a®xed number of days.
(Source: Kenneth Leong, ``The Right Approach,'' V@R: Understanding and Applying Value-at-Risk [New York, KPHGPublications, June 1996.])
CHAPTER 7 RISK ANALYSIS
365
but relatively off target at the trader's level: The ®rst one is aggregationand the second is its stochastic nature.The purpose of a risk analysis is highly dependent on the targetaudience, and for a trader, a risk report must focus absolutely andprimarily on the positions he or she is responsible for. In addition, arisk analysis should provide a clear view of each individual risk factor,such as stock prices, interest rates, and foreign exchange rates, becauseeach category calls upon different techniques and instruments. In otherwords, the level of details he or she is looking for is inadequate foranybody else, even sometimes his or her direct manager. V@R hasspeci®cally been designed with aggregation as a stated purpose,which creates a con¯ict with individual traders' needs.Beyond this problem, a more important ``¯aw'' in V@R methodology has to do with the nature of the reported result. A V@R calculationincorporates the randomness of the market because this is what itsusers need. The sole function of a trader, on the other hand, is to``manage'' this randomness because this is exactly how he or shemakes money. Therefore, he or she needs it stripped away, and he orshe will then make decisions based on this report and his or her ownintuition. A V@R report is only descriptive whereas a trader needs aprescriptive document, packaging information in a way suitable totrade on it.This is not to say that the two approaches are incompatible to thepoint where one represents the absolute truth and the other is plainlywrong. Incompatibilities do exist, but are dependent mainly on theaudience's needs and not on the informational content. Typically, forexample, an experienced trader is probably capable of estimating easilya V@R for the positions he or she knows and manages, which indicateshow close the two information sets are.It is important also to note that the use of different tools by different people leads to potential misunderstandings and con¯icts, whichconstitutes another reason to investigate these tools. Consider, forexample, a trader for whom risk limits are ®xed in V@R terms. Ifthis trader uses a partial derivative approach to monitoring risk, heor she faces a potential challenge in matching both methods andmaking the right decision at a given time.
CONCLUSION: THE LEPTOKURTIC CHALLENGERegardless of the method, risk is about predicting the unpredictable.And, as a general rule, the more unexpected the event, the larger the
366
FINANCIAL VALUATION AND RISK ANALYSIS
damages. The obvious example in the world of ®nancial markets is acrash, and 1987 is a perfect illustration. In terms of statistical relevance,October 1987 was a 20+ standard deviation event, which, assuming anormal distribution, has a very, very low probability. By comparison, aprobability of 1 occurrence in a 1,000 years is ``only'' 4.5 standarddeviations.29In fact, the recurrent occurrence of events that should statisticallynever happen is a well-known property of stock prices, and moregenerally of ®nancial assets. It is indeed the major ¯aw in the assumption that asset returns are normally distributed and one reason whyRiskMetrics has been consistently criticized over the years. Statisticallyspeaking, distributions with that characteristic are known as leptokurtic,which essentially means that they have tails fatter than those of anormal distribution. As an illustration, consider the case of Microsoft(MSFT). Figure 7-11 shows the histogram of returns from December 31,1993, to mid-May 2000.The kurtosis of the distribution is 6.12, way higher than that of anormal distribution, 3.30 The points on the left represent sharp overnight losses, with the minimum 15:60 percent on April 24, 2000. Thesecond worst is 14:47 percent, on April 3, 2000. On the other side, themaximum is 9.88 percent on December 15, 1999, followed by 9.68percent on April 18, 1997. At 3 standard deviations, the cumulativeprobability for MSFT is 0.437 percent, and the normal cumulative probability 0.106 percent. In other words, an unfavorable event on MSFThas 4.12 more chances (= 0.437 percent/0.106 percent) of happening onMSFT than on a normally distributed stock. Conversely, at +3 standarddeviations, MSFT has a score of 0.25 percent, compared to 0.11 percentfor a normal curve. Here the ``likelihood'' ratio is ``only'' 2.2.Although MSFT is a striking example, the leptokurtic challenge is amuch more general problem, arguably the most important in risk management. If market parameters follow stochastic processes relativelywell known, virtually all of them are likely to exhibit leptokurtic distributions, with the consequence that out-of-sample observations are areality, not a theoretical concept. Traders and risk managers shouldkeep this particular fact in mind when assessing risk. Managing riskon a daily basis is certainly a challenging task, but it pales compared tomanaging out-of-sample observations. In fact, many such realizations,29
At 252 open days a year, 1,000 years have 252,000 open days. 1 in 252,000 is0.000396825 percent, which in turn yields a Z score of 4:47 when plugged into areverse normal cumulative distribution.30The higher the kurtosis, the fatter the tails.
CHAPTER 7 RISK ANALYSIS367
Figure 7-11 Histogram, MSFT(Source: Compiled by author from Datastream data.)
368
FINANCIAL VALUATION AND RISK ANALYSIS
albeit less dramatic than 1987, have been responsible for ®nancial disasters, of which Barings is one example.31The same remarks in fact apply to a number of risks, and probablymost of them under one form or another. In general, unexpected eventstend to alter and damage established processes, with the consequenceof generating decisions that are suboptimal if not plainly wrong.Although rigorous analysis certainly helps, it is literally impossible toprepare for every contingency, and this is exactly where leptokurticevents tend to strike and make the most damage.
31
The Barings collapsed in 1995 from heavy lossesÐ$1.3 billionÐin its Singapore subsidiary. A single trader, Nick Leeson, had taken excessive positions on the Nikkei 225future in Osaka by forging and falsifying internal reporting documents. When theKobe earthquake struck, the sudden drop in the market generated huge losses forwhich he was unable to meet the margin requirements.
P ART IIICOMMON ARBITRAGESITUATIONS
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8I NDEX A RBITRAGE
INTRODUCTIONIndex arbitrage is the simultaneous sale and purchase of an indexfuture and of a portfolio replicating the same index. At a given time,an opportunity exists if the futures is different from its theoretical valuecomputed from a cost-of-carry model. Speci®cally, with the followingset of notations:. FTH , FM theoretical and market prices of the futures. I current value of the index
. Pi current price of the ith component of the index
. Wi , D de®nition of the index (weights and divisor). rF
zero-coupon market rate from today's settlement date to thefutures expiry date1. di future value of the accumulated dividends paid by the ith indexcomponent, as of the future expiry dateAn arbitrage situation in which we would buy the cash and sell thefutures can be illustrated as shown in Figure 8-1 with:PPWi  PiWi  diiid IFTH  I  1  rF  dDDThe difference bTH  FTH I  rF  I d is called the theoretical basis, orthe fair value of the futures. It indicates the amount by which the price
1
That is, from today's settlement date to the last settlement date of the futures contract.
371
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
372
COMMON ARBITRAGE SITUATIONS
FMbM
mispricing
FTH
bTI
Figure 8-1
Arbitrage Situation
of the futures should exceed the price of its underlying.2 The quantitybM  FM FTH is often called the market basis, and it indicates theamount by which the futures market price exceeds its theoreticalprice at a given time. Therefore, an opportunity exists when bM 6 0.If bM > 0, we buy a replicating portfolio and sell the futures, and viceversa if bM < 0.To understand why this is an opportunity, look at Figure 8-2 andrecall the reasoning we used to establish FTH . Assume we buy 1 indextoday (t0 ) at I, and sell 1 futures at FM . In order to cover the amount I,we borrow it at rF until expiry. On expiry (t1 ), the futures expires on its®nal settlement price (Special quotation, or SQ), and we sell the indexportfolio while paying back the amount I  1  rF ) as repayment for theloan. There is no treasury balance on t0 because we borrow exactlywhat we need, and the sale of the futures does not generate any cash¯owÐleaving aside margin calls for now. On t1 , the total cash ¯ow onexpiry is equal to:CF 
I  1  rF   FSQ
FSQ
FM   d  FM
FTH
The term I  1  rF ) is the payment of the loan including interest. The®rst FSQ is the sale of the index replicating portfolio at the SQ price. Thebracket (FSQ ± FM ) is the ¯ow generated by the expiry of the futures atFSQ because we were short. Finally, d is the amount of dividendreceived through the life of the position, on which we have receivedinterest day to day. In effect, the cash ¯ow on expiry is exactly equal to2
The theoretical basis is in general a positive number, but it can very well be negative.This is the case, for example, in Japan just before some ex-dates because the interestrate rF is smaller than the accumulated expected dividend d .
CHAPTER 8 INDEX ARBITRAGE
Toda y
t0, I
t1, FSQ
Expiry
rF
d
Figure 8-2
373
rd
Cash and Carry
the market basis at initiation of the position. If positive, this amount isexactly the pro®t generated by the arbitrage. Note that it is not conditional in any way, which is why it is considered risk free. The onlyapproximation is that we left margin calls aside temporarily, but wewill revisit this issue.3This presentation is all we need to understand the basics of indexarbitrage, but in practice, it is necessary to examine a number of peripheral issues that make its implementation less straightforward. Thisdetailed examination is exactly the purpose of the discussion below.After presenting all the tools and techniques independently in previousparts, we focus now on tying them together and try to complement thepresentation with as many practical examples as possible.
ABSOLUTE CONVERGENCEIndex arbitrage is based on an absolute convergence, and this propertyhas a number of consequences as we indicated very early. Recall theconvergence continuum, as shown in Table 8-1. Because the convergence of a futures contract with its underlying cash is the ``ultimate''convergence in terms of certainty, we do not need to spend a lot of timeon items such as valuation, convergence type, or convergence risk. Indoing so, we explicitly negate the possibility of a failure in the ®nancialsystem, which would be the only reason why the futures contract doesnot converge to its underlying cash price on expiryÐbecause this typeof event is extremely unlikely on any of the markets we consider.Once uncertainty about the convergence is cleared, we have to lookfor second-order risks. In this case, there are really three categories thatshould attract our attention: execution, hedge, and carry. Each of themneeds a careful discussion and has implications in a number of direc3
In addition, the stock position can be lent out partially or entirely, generating additional revenues. For simplicity, we leave this particular case out, but it would besimple to include in the overall valuation.
374
COMMON ARBITRAGE SITUATIONS
TABLE 8-1ConvergenceValuationConvergence typeConvergence riskExecution riskHedgeRisk of carryTime horizonWindow of opportunityPro®tabilityRobustnessImplications
Absolute
Hypothetical
TheoreticalPublicLow to nonexistentCriticalFixed quantityImportantKnown or boundedVery short10  1StrongPricing
Mark to marketHighly proprietaryCriticalLowFixed quantity or deltaLowUnknownLonger1  10WeakNone (Inef®cient market?)
tions. Before diving into the subject, however, let's take a somewhatbird-eye's-view perspective and examine the notions of time horizon,window of opportunity, and pro®tability.
OPPORTUNITIES AND PROFITABILITYOpportunities
As it appears, opportunities for an index arbitrageur seem relativelysimple to identify: They are exactly equal to the difference between thefutures market price and its fair value. Naturally a subsequent andimportant question is whether they can be captured, but if we simplychart the market basis, we should get a visualization of the marketintegration between cash and futures.Figures 8-3 through 8-7 present exactly this informationÐthemarket basisÐover a period of several years, at the close of eachmarket. To improve visibility, the actual basis has been averagedover 40 days and represented as a percentage of the underlyingindex value, and the chart overlays the 40-day moving average valueof the index as well.4
4
The fair value charted in the ®gures is always computed on the nearest futures contractÐthe ``front month''Ðand this may be inadequate just before an expiry when thefollowing futures contractÐthat is, the ``back month''Ðbecomes more liquid.However, this is entirely suf®cient for the purpose of the exercise, which is to get asense of the respective behaviors of each market.
CHAPTER 8 INDEX ARBITRAGE
375
S&P 500 Market Basis and Index Price(40-day moving average)0.15%
1,600S&P 500 40-day Moving Average Market BasisS&P 500 40-day Moving Average
0.10%
1,400
0.05%
1,200
0.00%
1,000
-0.05%
800
-0.10%
600
-0.15%Feb-95
400Aug-95
Feb-96
Jul-96
Jan-97
Jul-97
Dec-97
Jun-98
Dec-98
Jun-99
Nov-99
Figure 8-3 S&P 500 Market BasisNote: Fair value is estimated with a standard cost-of-carry model.(Source: Compiled by author from Datastream and Bloomberg data.)
FTSE Ma rket Basis and Index Price(40-day moving average)
0.30%
7,000FTSE 40-day Moving Average Market BasisFTSE 40-day Moving Average
6,500
0.20%6,0000.10%
5,5005,000
0.00%4,500-0.10%
4,0003,500
-0.20%3,000-0.30%Feb-95
2,500Aug-95
Feb-96
Jul-96
Jan-97
Jul-97
Dec-97
Jun-98
Dec-98
Figure 8-4 FTSE 100 Market BasisNote: Fair value is estimated with a standard cost-of-carry model.(Source: Compiled by author from Datastream and Bloomberg data.)
Jun-99
Nov-99
376
COMMON ARBITRAGE SITUATIONSNikkei 225 Market Basis and Index Price(40-day moving average)
0.45%
23,000
0.35%
22,00021,000
0.25%
20,000
0.15%
19,0000.05%18,000-0.05%17,000-0.15%
16,000
-0.25%
15,000NiKkei 225 40-day Moving Average Market Basis
-0.35%-0.45%Feb-95
14,000
Nikkei 225 40-day Moving Average
13,000Aug-95
Feb-96
Aug-96
Feb-97
Jul-97
Jan-98
Jul-98
Jan-99
Jul-99
Figure 8-5 Nikkei 255 Market BasisNote: Fair value is estimated with a standard cost-of-carry model.(Source: Compiled by author from Datastream and Bloomberg data.)
DAX 30 Market Basis and Index Price(40-day moving average)
0.10%
6,5006,000
0.00%5,500-0.10%
5,0004,500
-0.20%
4,000-0.30%
3,5003,000
-0.40%
2,500-0.50%
DAX 30 40-day Moving Average Market Basis
2,000
DAX 30 40-day Moving Average-0.60%Feb-95
1,500Aug-95
Feb-96
Jul-96
Jan-97
Jul-97
Jan-98
Jul-98
Dec-98
Figure 8-6 DAX 30 Market BasisNote: Fair value is estimated with a standard cost-of-carry model.(Source: Compiled by author from Datastream and Bloomberg data.)
Jun-99
Dec-99
CHAPTER 8 INDEX ARBITRAGE
377
CAC 40 Market Basis and Index Price(40-day moving average)
.20%
6,0005,500
.10%
5,000.00%4,500.10%
4,000
.20%
3,5003,000
.30%
2,500.40%2,000CAC 40 40-day Moving Average Market Basis
.50%
1,500
CAC 40 40-day Moving Average.60%Feb-95
1,000Aug-95
Feb-96
Aug-96
Jan-97
Jul-97
Jan-98
Jul-98
Jan-99
Jun-99
Dec-99
Figure 8-7 CAC 40 Market BasisNote: Fair value is estimated with a standard cost-of-carry model.(Source: Compiled by author from Datastream and Bloomberg data.)
Naturally these charts should be taken for what they areÐthat is,gross indications using rough estimates in terms of market parameters.Interest rates are LIBORs, and dividends are estimated from each indexdividend yield, with no real correction for actual ex-dates. In addition,dividends are taken as approximate amounts regardless of any withholding tax or tax credit, and we know that in European marketsnotably these notions tend to have a signi®cant impact on arbitrageursand eventually on the market fair value.As was already the case with interest rates, we de®nitively seedifferent behaviors expressed in these charts, and accordingly, individual arbitrageurs on these markets will have different strategies tocapture price discrepancies. Before we enter into details with respectto local particularities, the following two broad remarks should bemade.First, we are looking at the close, which is a very particular time ofthe trading day. It can be argued that because the market instantaneously closes at that time, arbitrage is extremely dif®cult if not impossible, and that is essentially the reason that the above charts showsizable price discrepancies recurrently. Indeed, in the case of the S&P500, the problem is even worse because the futures close 15 minutes
378
COMMON ARBITRAGE SITUATIONS
after the stocks; therefore, the chart in Figure 8-3 presents highly nonsimultaneous data. Although there is substance and truth in thesearguments, they do not explain why the market basis seems to followa cyclical pattern in every country, nor do they tell us why this pattern ofwaves seems to be centered approximately on the x axis, which isprecisely the no-arbitrage situation.5 The reasonable conclusion todraw from these pictures, despite the apparent low signi®cance ofthe data, is that the basis at close does capture information from themarket, in the form of a ``market sentiment.'' In a climate of pervasiveselling pressure, for example, it is reasonable to assume that thefutures, being more liquid, capture a larger share of that pressure,including at the close, which would essentially lead to a negative market basis at that time. This may seem like a purely speculative assumption; however, in practice, it is often the case that nervous markets tendto close with a negative basis and vice versa.6 In turn, if this phenomenon is strong enough to ®nd its way into the closing basis, it is notunreasonable to assume that it gets re¯ected somewhat in the intradaybasis, creating opportunities also during the day. This short discussionbrings to light a possible explanation for arbitrage opportunities in thepresence of an absolute convergence: positive or negative expectationsthat get re¯ected more heavily in futures because of their intrinsicnature.The second broad remark that needs to be made about these chartsconcerns their high level of legibility. Every single one of them containsa wealth of information about the underlying market, and each oneshould be carefully examined. The type of information extracted fromthe S&P 500 data has little in common with that obtained from lookingat the DAX picture, but both tell us a lot. For example, the magnitudeof the S&P 500 basis indicates that the U.S. market is by far the mostef®cient. Its deviation remains within 0:15 percent and 0:15 percentconsistently, while others oscillate within wider bounds. The Nikkei225 chart shows a remarkable relationship between the sign of the basisand the direction of the market, and this relationship could give an5
This last observation obviously does not appear to France and Germany because ofstrong dividend effects as we will see shortly. However, a wave pattern still developsin these markets.6Indeed, there seems to be academic evidence of that property, well known frommarket practitioners. See, for example, Nai-fu Chen, Charles J. Cuny, and Robert A.Haugen, ``Stock Volatility and the Levels of the Basis and Open Interest in FuturesContracts,'' Journal of Finance, 50, 1995, pp. 281±300. The article concludes that ``empirical evidence con®rms the model's prediction that increased volatility decreases thebasis and increases the open interest.''
CHAPTER 8 INDEX ARBITRAGE
379
indication as to whether the market basis can be used as an indicationof things to come.Figures 8-3 through 8-7 include ®ve years of data, but longer timeframes would give us an even better view of the underlying markets.As an illustration, consider the chart in Figure 8-8 showing the percent deviation from fair value from June 19827 to January 1999 for theS&P 500. Clearly, the basis reacted to periods of troubled markets like1987 and 1989, but more generally it is fair to say that the trend istoward lower and lower discrepancies. If we compared the two periods 1982 through 1989 to 1995 through 1999, it is apparent that themagnitude of the market basis in the latter period is considerably
Figure 8-8 S&P 500 LT Market Basis[Source: Jack C. Francis, William W. Toy, and J. Gregg Whittaker, The Handbook ofEquity Derivatives (New York: John Wiley & Sons, 2000.)]
7
June 1982 is the original date of listing of the S&P 500 contract on the ChicagoMercantile Exchange.
380
COMMON ARBITRAGE SITUATIONS
smaller than that of the previous one. There cannot be any doubt thatthis is the result of improved market ef®ciency and a better marketwide understanding of futures and their relationships to cash products.8United StatesThe case of the United States is particularly interesting because it servesas a benchmark of sorts. The fact that the basis at close oscillatesbetween 0:15 percent and +0.15 percent indicates that the market isprobably more mature and ef®cient than anywhere else. Even if making such a conclusion is relatively far-fetched from close-to-close data,which we know are biased because of nonsimultaneous prices, experience shows that this is indeed a fair description of the marketplace.Another interesting remark is the decorrelation between the sign ofthe basis and the direction of the market. We would expect the marketto be headed downward in periods of low negative basis, and conversely in periods of high positive basis. The reasons for this are not asmuch related to the activity of arbitrageurs as to the notion that arelative buying or selling pressure tends to appear more rapidly onfutures than on cash, creating punctual mispricings and driving thebasis one way or the other. The absence of such correlation, combinedwith the fact that the basis is evenly distributed above and below the xaxis, suggests that the market is ef®cient enough to be well arbitragedregardless of the overall market conditions. This last statement is con®rmed by trading data available from the NYSE. During the period ofJune 1998 to June 2000 the total volume dedicated to program tradingand index arbitrage was relatively steady beyond weekly variations,compared to the market performance or its basis.9 Figure 8-9 shows thevolume statistics, and it is relatively easy to compare it to the chart ofthe S&P 500 to come to this conclusion.Taken all together, these elements point toward a market moreef®cient than its peers, in which participants adapt extremely rapidlyto new market conditions without any visible effect on the basis, andfor which the basis is relatively low and stable in absolute terms. It is8
There is theoretical evidence as well that index arbitrage reduces basis volatility. See,for example, Praveen Kumar and Duane J. Seppi, ``Information and Index Arbitrage,''Journal of Business, 67, 1994, pp. 481±509.9The data are reported weekly by all members to the exchange. The index arbitragevolume is included in the total volume of program trading. For example, if programtrading is 20 percent and index arbitrage 5 percent, then 25 percent (= 5 percent/20percent) of all program trading transactions are index arbitrage.
CHAPTER 8 INDEX ARBITRAGE
381
Program Trading and Index Arbitrage as % of Total NYSE Volume
Figure 8-9 Program Trading(Source: New York Stock Exchange, www.nyse.com.)
relatively natural from there to conclude that the United States represents the ``ultimate'' phase of index arbitrage maturity. The basis variability there is not due to a lack of ef®ciency but rather to the intrinsiclevel of noise embedded in the structure of the market, and due to thedifference in liquidity and volatility between the futures and its underlying index. In other words, it is almost as if the market reached asteady state in which opportunities will persist inde®nitely, but con®ned to an extremely narrow range.JapanJapan is also an extremely interesting case because none of the remarkswe made for the United States truly applies. First, the magnitude of thebasis variations tend to be larger, almost twice as largeÐbetween 0.20percent and 0.30 percent. Experience shows indeed that index arbitrageis more pro®table in percentage terms in Japan than in the UnitedStates. More interestingly, Japan is the only market in our list havingexperienced repeated sharp downturns over the past few years, whichenables a more detailed examination of the correlation between basisand market performance. Indeed, Figure 8-5 shows a strong correlation, a rising market being accompanied almost always by a positivebasis and vice versa. This particular relationship could suggest that
382
COMMON ARBITRAGE SITUATIONS
arbitrageurs are driving the market; however, this is a grossly misstated conclusion because arbitrageurs are always hedged and havean equal nominal impact on both legsÐfor a net impact of zero. Infact, this property indicates that the ``expectation model''Ðthat is, thetendency of the basis to re¯ect market expectationsÐis stronger inJapan than anywhere else, a fact that is con®rmed easily by experience.United KingdomThe case of the United Kingdom represents a curiosity because itadopted an electronic platform in October 1997. Prior to that date,equities were traded in a dealer market, where no central electronicsystem was