<s> THE COMPLETEARBITRAGEDESKBOOK
Other Titles in the Irwin Library of Investment and FinanceQuantitative Business Valuationby Jay B. AbramsHigh-Yield Bondsby Theodore Barnhill, William Maxwell, and Mark ShenkmanPricing and Managing Exotic and Hybrid OptionsBy Vineer BhansaliConvertible Securitiesby John P. CalamosRisk Management and Financial Derivativesby Satyajit DasThe Handbook of Credit Derivativesby Jack Clark Francis, Joyce Frost, and J. Gregg WhittakerAsset Allocation, 2nd editionby Roger GibsonActive Portfolio Management, 2nd editionby Richard Grinold and Ronald KahnEquity Managementby Bruce Jacobs and Kenneth LevyThe Hedge Fund Handbookby Stefano LavinioImplementing Credit Derivativesby Israel NelkenPricing, Hedging, and Trading Exotic Optionsby Israel NelkenValuing a Business, 4th editionby Shannon Pratt, Robert Reilly, and Robert P. SchweihsValuing Small Business and Professional Practices, 3rd editionby Shannon Pratt, Robert Reilly, and Robert P. SchweihsThe Handbook of Advanced Business Valuationby Robert F. Reilly and Robert P. SchweihsValuing Intangible Assetsby Robert F. Reilly and Robert P. SchweihsManaging Financial Riskby Charles W. SmithsonGlobal Investment Risk ManagementBy Ezra Zask
THE COMPLETEARBITRAGEDESKBOOK
STEPHANE REVERRE
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CONTENTS
Spread and MidmarketNonstandard MaturityInterest Rate SwapsZero-Coupon Term StructuresForward Rate AgreementsSwap PricingShort-Term Interest Rate Futures ContractsForeign-Exchange Swaps and ForwardsFutures RevisitedConclusion
5
ix
175177177185194197202217221227
AGGREGATE INDICATORS: STOCK INDICES
229
IntroductionDe®nitionsMajor IndicesSponsorshipIndex Trading and Futures HedgingIndex ContinuityCorrelationsConclusion
229230234239244246262264
PART IIFINANCIAL VALUATION AND RISKANALYSIS
6
VALUATIONS
267
IntroductionDe®nitionsPrinciplesInstruments
267269269283
To Francoise and Ulyssefor their patience and support.To my parents and familyfor all they taught me.
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C ONTENTS
PREFACE
xiii
PART ITHEORY AND TOOLS
1
INTRODUCTION
3
What Is Arbitrage?ContextDifferent Types of ConvergenceArbitrage and PricingValuations: Mark to Market versus TheoreticalRiskConvergence ContinuumOther Examples
35112127294344
2
STOCKS
63
IntroductionDe®nitionsSettlementCross-Exchange ListingDividendsFundamentalsBorrowing and Lending of Securities
63646772768283
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viii
CONTENTS
TradingCapital Structure: Outstanding Number of Shares
3
9197
FUTURES AND RELATED INSTRUMENTS
107
IntroductionDe®nitionsClearinghouses, Margin Calls, and DepositsGeneral Margin Call FormulaCustomer Margin Calls Versus ExchangeMargin CallsStandard Portfolio Analysis of RiskSpecial Quotation Calculation and ConvergenceRevisitedFinal SettlementsExposure versus InvestmentCross ListingsTradingOther Forms of First-Order Exposure
107110117122
4
125126127130130132143146
MONEY MARKETS
151
IntroductionMoney Markets and Government InstitutionsInterbank Markets and Interest RatesCredit RiskOvernightTreasury Management versus Interest Rate ExposureLondon Inter-Bank Offered RateDomestic versus International RatesConventionsNotationsSettlement
151152153154155160161163168174175
x
CONTENTS
Costs and FeesCost of Capital
7
317320
RISK ANALYSIS
327
IntroductionRisk and Human PsychologyPrinciplesMarket RisksValue-at-RiskConclusion: The Leptokurtic Challenge
327328330340359365
PART IIICOMMON ARBITRAGE SITUATIONS
8
INDEX ARBITRAGE
371
IntroductionAbsolute ConvergenceOpportunities and Pro®tabilityRisksConclusionReferences
371373374391422422
9
RISK ARBITRAGE
423
IntroductionExplicit ConvergenceOpportunities and Pro®tabilityRisksIllustrationConclusion
423426427433449457
CONTENTS
10
xi
PAIR TRADING AND TECHNICAL TRADING
459
IntroductionSpeculative ArbitrageOpportunities and Pro®tabilityRisksIllustrationConclusion
459461463468474497
INDEX
499
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P REFACEThe ®rst and foremost question to ask about a new book, and one that Ihave been asked repeatedly, is ``Who is the target reader?'' In theparticular case at hand, the answer is relatively easy: Considering itsheavy algebraic content, this book is written for junior professionals oncapital markets, or nonprofessionals with a serious interest in WallStreet. Both groups will ®nd here the basic knowledge necessary tomake informed arbitrage decisions, at least in the world of equities.This precise focus does not mean that another public could not ®ndattractive substance in these lines. There is a lot to deal with, sometimeswith painful details, and a weakly motivated reader would probablysee less value overall. I have tried, however, to address this concern bykeeping some parts entirely algebra-free, for example, Part I. Meant tointroduce the subject and the products, it could very well be seen assuf®cient for anybody interested in getting a thorough introduction.Along the same lines, Part III is essentially a qualitative description ofwell-known arbitrage situations and could also be considered educational in its own right.In short, even if the mathematical treatment was a necessity toachieve the goal of a rigorous coverage, I hope to have done an adequate job at laying down much more than that, enough to keep even amildly technical reader interested.This hope is in fact rooted in what I believe makes this bookunique. I am not the ®rst one to write about equity products or arbitrage techniques. Indeed, arbitrage is the abc of modern trading, and assuch is present in one form or another in virtually all books on ®nancialmarkets. The originality here is the fact that the present work is entirelydedicated to arbitrage, precisely equity arbitrage. If I have found countless descriptions more or less theoretical of arbitrage on ®xed-incomeinstruments, I have found very little on the equity side. If there is anabundant literature on relative-value trading, notably risk arbitrage, Ifound very little on index arbitrage, and very little trying to tie togetherthe many strategies practiced throughout the marketplace. Therefore tothe best of my knowledge, there is no precedent in offering a comprexiii
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xiv
PREFACE
hensive view of equity arbitrage, including a theoretical presentation aswell as a detailed mathematical development.In addition, the treatment of such-and-such technique is generallyfocused on a single market, typically the United States. Naturally formany this is a natural consequence of the size and maturity of the U.S.®nancial markets, but this is hardly a satisfactory justi®cation becauseforeign markets, albeit smaller in size, would be far more interesting ifthey were indeed less mature speci®cally because trading would yieldhigher margins. Alternatively, one could argue that there is no need toexpand a given idea to foreign markets because if it works in one placeit probably works in others. Unfortunately this is not necessarily true. Ihave had the chance in my professional experience to be exposed toseveral of the largest markets, always in the context of an arbitrageactivity. I found the differences between these markets fascinating, andI realized progressively that even if the tools were essentially the same,local particularities were a de®ning factor in the existence and shape oftrading opportunities. For example, in European markets, ®scal arbitrage has traditionally been the bread and butter of index arbitrageurs,whereas opportunities of that type are almost unknown in Japan or theUnited States. Hence the second differentiating factor of this book: itsinternational scope. The point is naturally not to propose enoughdetails to enable the reader to trade indifferently in any of the marketscovered, but rather to offer insights from diverse backgrounds as a wayto uncover the universe of what is possible. For example, a situation inwhich interest rates diverge by up to 0.50 percent between domesticand off-shore institutions is extremely counterintuitive; still, this isexactly what happened in Japan repeatedly because of its domesticliquidity crisis.Finally I believe that the last important aspect of this work is anattempt to unify miscellaneous techniques under a generic umbrella.An arbitrage is a convergence, and that simple fact dictates the entireframework in which an opportunity should be evaluated and managed. I tried my best to develop that concept and the surroundingframework. And, as before, the purpose is not to answer all the questions, just the most important ones. The goal is to ``educate'' in thetraditional sense: In the end, the reader should be able to ask morequestions than those presented here and have a fair sense of where tolook for the answers.Note, however, that if I tried to include as much substance aspossible, there are many things that this book is not. For one, it isnot a trader's user's manual. In other words, nobody will gain fromthese lines a precious knowledge that could be turned into an immedi-
PREFACE
xv
ate pro®t. The trading ideas exposed are always well known and practiced actively. As I mention on many occasions, experience is the bestpredictor of success on a trading ¯oor, and you cannot learn about thatin a book. Another important point is that this book is far from exhaustive, even in its mathematical presentation. In the interest of space andtime I have left out numerous instruments, such as equity-linkedswaps. Along the same lines there are many variations and ¯avors ofarbitrage that I don't mention, and certainly some that I don't evenknow about. Again the idea is to present an introduction solid enoughthat readers will be able to form their own judgments in due course.Finally, I am not even sure this work can be considered a ``reference.''Despite the many concepts that have a general application, the vocabulary or speci®c treatment of certain ideas may differ, for example,from a highly specialized hedge fund with only a few traders to a hugetrading ¯oor in New York. After all, nobody has ever been able to tellwhat the fair price of a stock should be at a given time, so don't expecttoo many absolute truths on Wall Street in general.Given the objective of a broad-yet-rigorous presentation of equityarbitrage for a potential future practitioner, it was necessary to startfrom the very beginning on many fronts. Naturally, the notion ofarbitrage is de®ned and discussed in detail, but ®nancial instrumentsused in the implementation of a position are ``dissected'' as well, eventhose traditionally considered simple, like stocks. The reason for thatchoice is based on my experience as a junior trader. Everybody knowswhat a stock is, but few people really understand the intricacies of®scal arbitrage, which is nothing more than a simple stock loan ifyou think about it. It took me a few months to get a sense of whatmy job would be as an arbitrageur, and years to painfully master thezillions of small details. My intention is to bring to light enough ofthese details that a reader will save considerable time in climbing theexperience curve.Part II of the book is built around the same idea. Valuation and riskanalysis are absolutely critical in the life and success of a professionaltrader. Too often are these performed by a beginner without a clearoverview of the purpose they serve and of their underlying principles.The emphasis here, therefore, is on those goals and principles, with thedevelopment of the algebra as an application exercise. Finally Part III ismeant to bring all the concepts together. Through the examination ofactual examples and practical situations, the goal is to achieve an integrated view of arbitrage opportunities, precisely as a trader would seethem.
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P ART ITHEORY AND TOOLS
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1I NTRODUCTION
WHAT IS ARBITRAGE?Arbitrage is ``the simultaneous purchase and sale of the same, or essentially similar, security in two different markets for advantageouslydifferent prices.''1 The word should be read in a very general sensebecause both legs of the arbitrage can be a number of securities orderivatives.The de®nition essentially underlines two structural features ofthese transactions. First, the operations on the different instrumentsmust be simultaneous. This characteristic is a direct consequence ofthe fact that arbitrage opportunities must be risk-free and are consequently short-lived. Secondly, the securities should be the same oressentially similar. If we broadened the scope, we would include securities that are not actually similar but are believed to behave in a verysimilar way. This similarity in essence is an insurance contract, and byconstruction, it eliminates all and every risk beyond immediate execution. This idea is commonly described as the law of one price, and it is1
Zivi Bodie, Alex Kane, and Alan J. Marcus, Investments, 4th ed. (New York: Irwin/McGraw-Hill, 1999), p. 307. The common term arbitrage is in fact used in many moresituations. See, for example, Jochen E. M. Wilhelm, Arbitrage Theory (New York:Springer-Verlag, 1985), which makes a distinction between arbitration (``the searchfor the lowest cost in achieving a certain intended ®nancial position''), spreads (simultaneous sale and purchase, i.e., our arbitrage), and free lunches ('`turning from onecombination of assets to another one which is equivalent but has a lower marketprice''). In this terminology, we are primarily concerned with spreads.
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4
THEORY AND TOOLS
used extensively in the context of derivative pricing: If two portfolioshave the same payoff at maturity, their price should be equal today,and if they are not, they can be expected to converge back towardequality.Therefore, a close examination of arbitrage should focusprimarily on these two aspects, execution and convergence.Understanding how we can execute requires an examination of whyopportunities appear in the ®rst place. In the context of mature ®nancial markets, ef®ciency is a given, and conventional wisdom statesexplicitly that arbitrage opportunities should not exist. Naturally markets would not be ef®cient without arbitrageurs to make them ef®cient,but we need to understand speci®cally how discrepancies occur. Inaddition, the convergence issue requires us to examine the notion ofmaturity: If the securities are actually fairly priced and similar, weshould be able to exit from the arbitrage at some point in the future.This may be a knownÐfor example, the expiry of a futuresÐorunknown date, but at that time both legs should come together. Insome cases the convergence is embedded in the systemÐa futurescontract, for exampleÐbut in many instances it is more assumedthan it is certain. The remote chance that it does not materialize is byfar the biggest risk faced by all arbitrageurs. Finally, we need to consider the implications of safely carrying this position for a period oftime that is typically uncertain. In some instances we know an upperbound for the time to exitÐderivative expiry, merger effective dateÐbut we do not know if an opportunity will not allow us, or an unexpected event force us, to exit before.In fact, the convergence problem entirely dictates the appropriatemethodology to evaluate and manage arbitrage positions, and that isprecisely what we will try to demonstrate. All risks are subordinated toit, and by nature this is the only one we cannot and do not want tohedge. For example, consider the case of index arbitrage: The convergence could not be more certain, and the arbitrageur's risk management is then focused on ``secondary'' risksÐtypically interest rates,dividends, and so on. At the other end of the spectrum, a pair tradertakes a position on stocks under a fairly optimistic assumption thattheir prices will remain highly correlated in the future. In that caseinterest rates and dividends do not matter much; pro®t and losseshave to be conservatively evaluated based on market prices as opposedto a theoretical relationship, and the primary measure of risk is thebehavior of the spread between the companies considered.Consequently the natural path to follow is to start with a closerexamination of the convergence issue. From there we will examine the
CHAPTER 1 INTRODUCTION
5
hypothesis of market ef®ciency and its implications, and we will propose a certain number of necessary conclusions about valuation andrisk, developed in Part 2.
CONTEXTMoney Managers versus Proprietary Traders
An essential prerequisite for this discussion is that it occur within thecontext of professional trading. We want to take the perspective of aproprietary trader in a large ®nancial institution, typically the tradingsubsidiary of a large bank.2 We cannot emphasize this particularityenough because it has fundamental implications in several respects:incentives and performance evaluation, information ¯ows, capitalavailability, and ®nancing costs.Proprietary traders are entrusted with money from their employerand allowed to take de®ned risks in order to generate as much pro®t aspossible. Their compensation packages are generally indexed verydirectly to their performance, usually after costs. The fact that thecapital used is proprietary means that this money is not an investmentfrom a set of investors looking for the best use of their excess resources.It is borrowed by the trader using the balance sheet of the institution.To understand the implications of this situation, consider in contrast atypical investment manager at a mutual fund, insurance company, orpension fund. The fund is an investment vehicle with a clear strategy.Investors hope for and expect a return, but these expectations do nothave any cost per se. Obviously, poor performance will make it harderto attract new clients and may eventually decrease assets under management, but the manager does not have to actually pay to attractfundsÐleaving marketing costs aside. A proprietary trader, on theother hand, incurs ®nancing charges as a true cost. If both the traderand fund manager buy a stock, hold it for six months and sell it at a2
In the United States, until recently, the 1933 Glass-Steagall Act strictly restricted thetype of activities that banks, securities ®rms, and insurance companies might engagein. As a result, banks used to set up subsidiaries to engage in trading activities. Overthe past few years, however, legislators have signi®cantly weakened the act's originalprovisions. See, Bernard Shull, ``The Separation of Banking and Commerce in theUnited States: An Examination of Principal Issues,'' for example, Financial Markets,Institutions and Instruments, vol. 8, no. 3, New York University Salomon Center, 1999.A similar regulation exists in Japan due to the heavy in¯uence of the post±World WarII U.S. regulators, but there is not one in Europe.
6
THEORY AND TOOLS
price equal to the entry price, the manager exhibits only a mediocreperformanceÐ``not my fault because the market did not move''Ðwhile the trader would show an actual loss equal to the ®nancingcost over six months.In fact, the cost is higher because the ®nancial institution carrying the borrowing on its balance sheet puts a fraction of its capitalat risk. International regulations and banking practices require banksto match part of their risky assets with equity to protect creditors,that is, depositors. For every $1 borrowed, the bank has to allocate,for example, $0.10 of its tier-one capital, which will not be availableto back any other opportunity. As a result of this commitment, thebank will ask the trader for a compensation commensurate with therisk taken and with its cost of equity. Say that this return is 15percent. Then $0.10 of capital must return $0.015 to pay for thecapital consumption, in addition to the ®nancing charge. Overall,the interest rate will be increased by 1.5 percent on the original$1. This is a complex description of a very intuitive fact: As a risktaker, the trader is required to generate returns above the risk-freerate.3 By contrast, investment companies are also required to maintain a capital base, but the protection against market movement isnot a concern anymore because investors are using such vehiclesexplicitly for the purpose of accessing the market.4One of the most fundamental differences between traders andmoney managers is in fact that managers are usually benchmarked.Consider again that both of them buy and hold a stock for six months.At this point they sell it at a loss, say, 5 percent. If the broader marketlost 5 percent over the same period, the fund manager has fared honorablyÐ''not my fault, in line with the market.'' On the other side, thetrader shows a loss of 5 percent, no matter what. Schematically,because they are benchmarked, money managers do not have anyexposure to broad market movements. Naturally this is a very shortsighted view because it does not apply to all fund managers; neitherdoes the opposite describe all traders. Hedge funds speci®cally arenotable exceptions: Because of their particular nature, most are onlyremotely benchmarked and thus carry a full market risk. However, for3
This discussion is an introduction to the notion of cost of capital and solvency ratios.Both will be discussed in more detail in Part 2.4Indeed, for investment companies the concerns are of two types: disclosure, toensure that fund managers are following their mandate and maintaining appropriaterisk levels, and protection of assets, against fraudulent activity and misappropriation.
CHAPTER 1 INTRODUCTION
7
the purpose of our introduction, this schematic description captures theessence of the issues at hand.An interesting corollary of this difference is the fact that traders areusually marked to market systematically every day, while money managers may not be subject to such stringent reporting requirements. Ofcourse, virtually all fund managers compute and publish a net assetvalue every day, but full disclosure to investors in most cases isrequired only periodicallyÐfor example, every quarter for mutualfunds. By contrast, trading positions are reviewed much more frequently by risk managers and head traders, with the consequencethat a particular strategy has to be justi®ed sooner if it takes too longto turn a pro®t. In addition, because in many cases marketing materialfor money managers is based on month-end data, it is easier for themto use window dressing to improve performance on paper.5The second set of structural differences materializes with regardto information ¯ows. For obvious reasons, professional traders havebetter access to research and real-time data than most investors, andthat applies to some extent to institutional investors as well. Forthem, the costs related to information systems are directly chargedagainst a fund's return. By contrast, on a trading ¯oor, it is mucheasier for traders to get sophisticated information because of economies of scale in disseminating this information throughout the entire®rm. For example, all major players maintain a quantitative andresearch department whose sole function is to crunch numbers continuously with a decent amount of brainÐand computingÐpower.These resources are naturally very expensive and not available toall.6Finally, probably the most important structural difference is theavailability of capital and its cost. As we will see, it is not uncommonfor arbitrage positions, particularly index arbitrage, to reach severalbillion dollars in nominal. These amounts can be borrowed in minutesat consistent ratesÐtypically the LIBOR7 with no dif®culty whatsoeverfor a trader in a large bank. This ¯exibility is critical in scaling positionsand risks in relation to market opportunities, and it is available only tothe largest names and the most creditworthy institutions.5
Window dressing refers to arti®cial trading activity designed to move prices of certainsecurities ahead of a particularly important date, typically month end or year end.6Naturally this presentation is a simpli®cation. Large institutional investors have adequate resources to compete with large trading ®rms, but it remains true that theunderlying economic model and the ability to absorb costs are not the same.7London Interbank Offered Rate. See Chapter 4 for a complete description.
8
THEORY AND TOOLS
For all these reasons, arbitrage tends to be practiced on a large scaleby a few quali®ed professionals, typically hedge fund managers andproprietary traders. It is generally beyond the scope of most otherinstitutional investors, and even if it were not, the structural featureswe listed above would virtually prevent them from competing withmore specialized players. We do not mean to imply that it is the secretgarden of a select few, but that only a limited number of people in thebroad population of investors have the resources to aggressively andsystematically pursue arbitrage opportunities.8 Index arbitrage is atypical example: Optimistically, there are no more than a dozen tradersactively involved in it on any major market.
Arbitrage and Competitive Advantage
If indeed arbitrageurs are relatively few in number with unlimitedaccess to capital, we need to ask what type of competitive advantageis needed to succeed.The ®rst one to spot an opportunity is probably the only one topro®t signi®cantly from it; therefore, competitive advantage here meansprimarily being the fastest to push the button. Consider, for example,that all arbitrageurs agree on what is an opportunity and what is not,as wouldÐalmostÐbe the case for an index arbitrage or risk arbitrage situation. Would all of them rush to trade at the same time? Inan ideal world where taxes and credit risks do not exist, the answerto this question is most probably yes. In reality, because of differencesin tax treatment or costs related to poor credit ratings, some arbitrageurs appear to enjoy structural advantages. For example, taxeswould change the amount of expected net dividends over a futureperiod, and credit ratings would change the ®nancing rate until thenext futures expiry, both of them affecting the fair value of an indexarbitrageur. The example of taxes is in fact very meaningful because itis truly discriminative among market participants. In Japan, for example, domestic and offshore entities have different transaction and
8
Indeed, it was for a long time truly performed by a select few, and thus it was a wellkept secret. See, for example, the description of arbitrageurs in Risk Arbitrage II fromGuy P. Waser-Pratte (Salomon Brothers Center for the Study of Financial Institutions,Graduate School of Business Administration, New York University, 1982): a ``cliquishband of specialists.'' The 1987 crash and the resulting bad press for proprietary tradersin general forced numerous institutions out of these strategies. In addition, the reporting requirements instituted afterward called for much more transparency and eventually brought to light what had been thus far a restricted gentlemen's club.
CHAPTER 1 INTRODUCTION
9
dividend taxes, which introduces a structural bias in trading parameters.However, structural differences of this nature are usually fairlyeasy to bypass because large institutions have the ®nancial andoperational ability to converge toward the optimal structureÐlegaland otherwiseÐto take advantage of market opportunities. Thusarbitrageurs in our context usually stand on common ground, andthese biases become minor. In addition, most of the time such differences are actual impediments to tradingÐtransaction or ®nancingcosts, for example. Therefore, their net effect is to eliminate playersfrom the ®eld while those remaining do not have any long-standingcompetitive advantage relative to their peers.If we assume then that structural differences cannot