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DERIVATIVES DE RI V A T IV ES

PRINCIPLES and PRACTICE PRINCIPLES and PRACTICE

Rangarajan K. Sundaram

Sanjiv R. Das

Derivatives: Principlesand Practice

StephenA.Ross

FrancoModiglianiProfessorofFinanceandEconomics SloanSchoolofManagement MassachusettsInstituteofTechnology ConsultingEditor

FINANCIALMANAGEMENT

Block,Hirt,andDanielsen FoundationsofFinancialManagement FifteenthEdition

Brealey,Myers,andAllen PrinciplesofCorporateFinance EleventhEdition

Brealey,Myers,andAllen PrinciplesofCorporateFinance,Concise SecondEdition

Brealey,Myers,andMarcus FundamentalsofCorporateFinance EighthEdition

Brooks FinGameOnline5.0 Bruner CaseStudiesinFinance:Managingfor CorporateValueCreation SeventhEdition

Cornett,Adair,andNofsinger Finance:ApplicationsandTheory ThirdEdition

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DeMello CasesinFinance SecondEdition

Grinblatt(editor)

StephenA.Ross,Mentor:Influence throughGenerations

GrinblattandTitman FinancialMarketsandCorporate Strategy SecondEdition

Higgins AnalysisforFinancialManagement EleventhEdition

Kellison TheoryofInterest ThirdEdition

Ross,Westerfield,andJaffe CorporateFinance TenthEdition

Ross,Westerfield,Jaffe,andJordan CorporateFinance:CorePrinciples andApplications FourthEdition

Ross,Westerfield,andJordan EssentialsofCorporateFinance EighthEdition

Ross,Westerfield,andJordan FundamentalsofCorporateFinance EleventhEdition

Shefrin

BehavioralCorporateFinance:Decisions thatCreateValue FirstEdition

White FinancialAnalysiswithanElectronic Calculator SixthEdition

INVESTMENTS

Bodie,Kane,andMarcus EssentialsofInvestments NinthEdition

Bodie,Kane,andMarcus Investments TenthEdition HirtandBlock FundamentalsofInvestmentManagement TenthEdition

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RoseandHudgins BankManagementandFinancialServices NinthEdition

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SaundersandCornett FinancialInstitutionsManagement: ARiskManagementApproach EighthEdition

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INTERNATIONALFINANCE

EunandResnick InternationalFinancialManagement SeventhEdition

REALESTATE

BrueggemanandFisher RealEstateFinanceandInvestments FourteenthEdition

LingandArcher RealEstatePrinciples:AValueApproach FourthEdition

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Allen,Melone,Rosenbloom,andMahoney RetirementPlans:401(k)s,IRAs,and OtherDeferredCompensation Approaches EleventhEdition Altfest PersonalFinancialPlanning FirstEdition

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Kapoor,Dlabay,andHughes PersonalFinance EleventhEdition

WalkerandWalker PersonalFinance:BuildingYourFuture FirstEdition

Derivatives: Principlesand Practice

SecondEdition

RangarajanK.Sundaram

SternSchoolofBusiness

NewYorkUniversity

NewYork,NY10012

SanjivR.Das

LeaveySchoolofBusiness

SantaClaraUniversity

SantaClara,CA95053

DERIVATIVES:PRINCIPLESANDPRACTICE,SECONDEDITION

PublishedbyMcGraw-HillEducation,2PennPlaza,NewYork,NY10121.Copyright©2016byMcGraw-Hill Education.Allrightsreserved.PrintedintheUnitedStatesofAmerica.Previousedition©2011.Nopartofthis publicationmaybereproducedordistributedinanyformorbyanymeans,orstoredinadatabaseorretrieval system,withoutthepriorwrittenconsentofMcGraw-HillEducation,including,butnotlimitedto,inany networkorotherelectronicstorageortransmission,orbroadcastfordistancelearning.

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PARTTHREE

MarketModels714

BibliographyB-1

IndexI-1

ThefollowingWebchaptersare availableatwww.mhhe.com/sd2e:

PARTSIX

Computation1

35 DerivativePricingwithFinite Differencing3

36 DerivativePricingwithMonteCarlo Simulation23

37 Using Octave 45

Contents

AuthorBiographiesxv

Prefacexvi

Acknowledgmentsxxi

Chapter1

Introduction1

1.1 ForwardandFuturesContracts5

1.2 Options9

1.3 Swaps10

1.4 UsingDerivatives:SomeComments12

1.5 TheStructureofthisBook16

1.6 Exercises17

PARTONE

FuturesandForwards19

Chapter2

FuturesMarkets21

2.1 Introduction21

2.2 TheFunctioningofFuturesExchanges23

2.3 TheStandardizationofFuturesContracts32

2.4 ClosingOutPositions35

2.5 MarginRequirementsandDefaultRisk37

2.6 CaseStudiesinFuturesMarkets40

2.7 Exercises55

Appendix2A FuturesTradingandUSRegulation: ABriefHistory59

Appendix2B Contango,Backwardation,and RolloverCashFlows62

Chapter3

PricingForwardsandFuturesI:TheBasic Theory63

3.1 Introduction63

3.2 PricingForwardsbyReplication64

3.3 Examples66

3.4 ForwardPricingonCurrenciesandRelated Assets69

3.5 Forward-RateAgreements72

3.6 ConceptCheck72

3.7 TheMarked-to-MarketValueofaForward Contract73

3.8 FuturesPrices75

3.9 Exercises77

Appendix3A CompoundingFrequency82

Appendix3B ForwardandFuturesPriceswith ConstantInterestRates84

Appendix3C RollingOverFuturesContracts86

Chapter4

PricingForwardsandFuturesII:Building ontheFoundations88

4.1 Introduction88

4.2 FromTheorytoReality88

4.3 TheImpliedRepoRate92

4.4 TransactionsCosts95

4.5 ForwardPricesandFutureSpotPrices96

4.6 IndexArbitrage97

4.7 Exercises100

Appendix4A ForwardPriceswithConvenience Yields103

Chapter5

HedgingwithFuturesandForwards104

5.1 Introduction104

5.2 AGuidetotheMainResults106

5.3 TheCashFlowfromaHedgedPosition107

5.4 TheCaseofNoBasisRisk108

5.5 TheMinimum-VarianceHedgeRatio109

5.6 Examples112

5.7 Implementation114

5.8 FurtherIssuesinImplementation115

5.9 IndexFuturesandChangingEquityRisk117

5.10 Fixed-IncomeFuturesandDuration-Based Hedging118

5.11 Exercises119

Appendix5A DerivationoftheOptimalTailed HedgeRatio h ∗∗ 124

Chapter6

Interest-RateForwardsandFutures126

6.1 Introduction126

6.2 EurodollarsandLiborRates126

6.3 Forward-RateAgreements127

6.4 EurodollarFutures133

6.5 TreasuryBondFutures140

6.6 TreasuryNoteFutures144

6.7 TreasuryBillFutures144

6.8 Duration-BasedHedging144

6.9 Exercises147

Appendix6A PVBP-BasedHedgingUsing EurodollarFutures151

Appendix6B CalculatingtheConversion Factor152

Appendix6C DurationasaSensitivity Measure153

Appendix6D TheDurationofaFutures Contract154

PARTTWO

Options155

Chapter7 OptionsMarkets157

7.1 Introduction157

7.2 DefinitionsandTerminology157

7.3 OptionsasFinancialInsurance158

7.4 NakedOptionPositions160

7.5 OptionsasViewsonMarketDirection andVolatility164

7.6 Exercises167

Appendix7A OptionsMarkets169

Chapter8

Options:PayoffsandTrading Strategies173

8.1 Introduction173

8.2 TradingStrategiesI:CoveredCallsand ProtectivePuts173

8.3 TradingStrategiesII:Spreads177

8.4 TradingStrategiesIII:Combinations185

8.5 TradingStrategiesIV:OtherStrategies188

8.6 WhichStrategiesAretheMostWidely Used?191

8.7 TheBaringsCase192

8.8 Exercises195

Appendix8A AsymmetricButterfly Spreads198

Chapter9

No-ArbitrageRestrictionson OptionPrices199

9.1 Introduction199

9.2 MotivatingExamples199

9.3 NotationandOtherPreliminaries201

9.4 MaximumandMinimumPricesfor Options202

9.5 TheInsuranceValueofanOption207

9.6 OptionPricesandContractParameters208

9.7 NumericalExamples211

9.8 Exercises213

Chapter10

EarlyExerciseandPut-CallParity216

10.1 Introduction216

10.2 ADecompositionofOptionPrices216

10.3 TheOptimalityofEarlyExercise219

10.4 Put-CallParity223

10.5 Exercises229

Chapter11

OptionPricing:AFirstPass231

11.1 Overview231

11.2 TheBinomialModel232

11.3 PricingbyReplicationinaOne-Period BinomialModel234

11.4 Comments238

11.5 RisklessHedgePortfolios240

11.6 PricingUsingRisk-Neutral Probabilities240

11.7 TheOne-PeriodModelinGeneral Notation244

11.8 TheDeltaofanOption245

11.9 AnApplication:PortfolioInsurance249 11.10 Exercises251

Appendix11A RisklessHedgePortfolios andOptionPricing255

Appendix11B Risk-NeutralProbabilities andArrowSecurityPrices256

Appendix11C TheRisk-NeutralProbability, No-Arbitrage,andMarket Completeness257

Appendix11D EquivalentMartingale Measures260

Chapter12

BinomialOptionPricing261

12.1 Introduction261

12.2 TheTwo-PeriodBinomialTree263

12.3 PricingTwo-PeriodEuropeanOptions264

12.4 EuropeanOptionPricinginGeneral n -Period Trees271

12.5 PricingAmericanOptions:Preliminary Comments271

12.6 AmericanPutsonNon-Dividend-Paying Stocks272

12.7 CashDividendsintheBinomialTree274

12.8 AnAlternativeApproachtoCash Dividends278

12.9 DividendYieldsinBinomialTrees282

12.10 Exercises284

Appendix12A AGeneralRepresentationof EuropeanOptionPrices287

Chapter13

ImplementingBinomialModels290

13.1 Introduction290

13.2 TheLognormalDistribution291

13.3 BinomialApproximationsofthe Lognormal295

13.4 ComputerImplementationoftheBinomial Model299

13.5 Exercises304

Appendix13A EstimatingHistorical Volatility307

Chapter14

TheBlack-ScholesModel309

14.1 Introduction309

14.2 OptionPricingintheBlack-Scholes Setting311

14.3 RemarksontheFormula315

14.4 WorkingwiththeFormulaeI:PlottingOption Prices315

14.5 WorkingwiththeFormulaeII:Algebraic Manipulation317

14.6 DividendsintheBlack-ScholesModel321

14.7 OptionsonIndices,Currencies, andFutures326

14.8 TestingtheBlack-ScholesModel:Implied Volatility329

14.9 TheVIXandItsDerivatives334

14.10 Exercises336

Appendix14A FurtherPropertiesofthe Black-ScholesDelta340

Appendix14B VarianceandVolatilitySwaps341

Chapter15

TheMathematicsofBlack-Scholes346

15.1 Introduction346

15.2 GeometricBrownianMotionDefined346

15.3 TheBlack-ScholesFormulavia Replication350

15.4 TheBlack-ScholesFormulaviaRisk-Neutral Pricing353

15.5 TheBlack-ScholesFormulaviaCAPM356

15.6 Exercises357

Chapter16

OptionsModeling: BeyondBlack-Scholes359

16.1 Introduction359

16.2 Jump-DiffusionModels360

16.3 StochasticVolatility370

16.4 GARCHModels376

16.5 OtherApproaches380

16.6 ImpliedBinomialTrees/LocalVolatility Models381

16.7 Summary391

16.8 Exercises391

Appendix16A ProgramCodeforJumpDiffusions395

Appendix16B ProgramCodeforaStochastic VolatilityModel396

Appendix16C HeuristicCommentsonOption PricingunderStochastic Volatility

Seeonlineatwww.mhhe.com/sd2e Appendix16D ProgramCodeforSimulating GARCHStockPrices Distributions399

Appendix16E LocalVolatilityModels:TheFourth PeriodoftheExample

Seeonlineatwww.mhhe.com/sd2e

Chapter17

SensitivityAnalysis:TheOption “Greeks”401

17.1 Introduction401

17.2 InterpretingtheGreeks:ASnapshot View401

17.3 TheOptionDelta405

17.4 TheOptionGamma409

17.5 TheOptionTheta415

17.6 TheOptionVega420

17.7 TheOptionRho423

17.8 PortfolioGreeks426

17.9 Exercises429

Appendix17A DerivingtheBlack-Scholes OptionGreeks433

Chapter18

ExoticOptionsI:Path-Independent Options437

18.1 Introduction437

18.2 ForwardStartOptions439

18.3 Binary/DigitalOptions442

18.4 ChooserOptions447

18.5 CompoundOptions450

18.6 ExchangeOptions455

18.7 QuantoOptions456

18.8 VariantsontheExchange OptionTheme458

18.9 Exercises462

Chapter19

ExoticOptionsII:Path-Dependent Options467

19.1 Path-DependentExotic Options467

19.2 BarrierOptions467

19.3 AsianOptions476

19.4 LookbackOptions482

19.5 Cliquets485

19.6 ShoutOptions487

19.7 Exercises489

Appendix19A BarrierOptionPricing Formulae493

Chapter20 Value-at-Risk495

20.1 Introduction495

20.2 Value-at-Risk495

20.3 RiskDecomposition502

20.4 CoherentRiskMeasures508

20.5 Exercises512

Chapter21

ConvertibleBonds516

21.1 Introduction516

21.2 ConvertibleBondTerminology516

21.3 MainFeaturesofConvertibleBonds517

21.4 BreakevenAnalysis521

21.5 PricingConvertibles:AFirstPass522

21.6 IncorporatingCreditRisk528

21.7 ConvertibleGreeks533

21.8 ConvertibleArbitrage540

21.9 Summary541

21.10 Exercises542

Appendix21A Octave CodefortheBlended DiscountRateValuationTree544 Appendix21B Octave CodefortheSimplified Das-SundaramModel545

Chapter22 RealOptions547

22.1 Introduction547

22.2 PreliminaryAnalysisandExamples549

22.3 ARealOptions“CaseStudy”553

22.4 CreatingtheStateSpace559

22.5 ApplicationsofRealOptions562

22.6 Summary563

22.7 Exercises563

PARTTHREE Swaps567

Chapter23

InterestRateSwapsandFloating-Rate Products569

23.1 Introduction569

23.2 Floating-RateNotes569

23.3 InterestRateSwaps573

23.4 UsesofSwaps574

23.5 SwapPayoffs577

23.6 ValuingandPricingSwaps580

23.7 ExtendingthePricingArguments586

23.8 CaseStudy:TheProcter&Gamble–Bankers Trust“5/30”Swap591

23.9 CaseStudy:ALong-TermCapital Management“ConvergenceTrade”595

23.10 CreditRiskandCreditExposure597

23.11 HedgingSwaps598

23.12 Caps,Floors,andSwaptions600

23.13 TheBlackModelforPricingCaps,Floors, andSwaptions605

23.14 Summary610

23.15 Exercises610

Chapter24

EquitySwaps614

24.1 Introduction614

24.2 UsesofEquitySwaps615

24.3 PayoffsfromEquitySwaps617

24.4 ValuationandPricingofEquitySwaps623

24.5 Summary629

24.6 Exercises629

Chapter25 CurrencyandCommoditySwaps632

25.1 Introduction632

25.2 CurrencySwaps632

25.3 CommoditySwaps643

25.4 Summary647

25.5 Exercises647

PARTFOUR

InterestRateModeling651

Chapter26

TheTermStructureofInterestRates: Concepts653

26.1 Introduction653

26.2 TheYield-to-Maturity653

26.3 TheTermStructureofInterestRates655

26.4 DiscountFunctions656

26.5 Zero-CouponRates657

26.6 ForwardRates658

26.7 Yield-to-Maturity,Zero-CouponRates,and ForwardRates660

26.8 ConstructingtheYield-to-MaturityCurve:An EmpiricalIllustration661

26.9 Summary665

26.10 Exercises665 Appendix26A TheRawYTMData668

Chapter27

EstimatingtheYieldCurve671

27.1 Introduction671

27.2 Bootstrapping671

27.3 Splines673

27.4 PolynomialSplines674

27.5 ExponentialSplines677

27.6 ImplementationIssueswithSplines678

27.7 TheNelson-Siegel-SvenssonApproach678

27.8 Summary680

27.9 Exercises680

Appendix27A BootstrappingbyMatrix Inversion684

Appendix27B ImplementationwithExponential Splines685

Chapter28

ModelingTerm-StructureMovements688

28.1 Introduction688

28.2 Interest-RateModelingversusEquity Modeling688

28.3 ArbitrageViolations:ASimple Example689

28.4 “No-Arbitrage”and“Equilibrium” Models691

28.5 Summary694

28.6 Exercises695

Chapter29

FactorModelsoftheTermStructure697

29.1 Overview697

29.2 TheBlack-Derman-ToyModel Seeonlineatwww.mhhe.com/sd2e

29.3 TheHo-LeeModel Seeonlineatwww.mhhe.com/sd2e

29.4 One-FactorModels698

29.5 MultifactorModels704

29.6 AffineFactorModels706

29.7 Summary709

29.8 Exercises709

Appendix29A DerivingtheFundamentalPDE inFactorModels712

Chapter30

TheHeath-Jarrow-MortonandLibor MarketModels714

30.1 Overview714

30.2 TheHJMFramework:Preliminary Comments714

30.3 AOne-FactorHJMModel716

30.4 ATwo-FactorHJMSetting725

30.5 TheHJMRisk-NeutralDrifts:AnAlgebraic Derivation729

30.6 LiborMarketModels732

30.7 MathematicalExcursion:Martingales733

30.8 LiborRates:Notation734

30.9 Risk-NeutralPricingintheLMM736

30.10 SimulationoftheMarketModel740

30.11 Calibration740

30.12 SwapMarketModels741

30.13 Swaptions743

30.14 Summary744

30.15 Exercises744

Appendix30A Risk-NeutralDrifts andVolatilitiesinHJM748

PARTFIVE

CreditRisk753

Chapter31

CreditDerivativeProducts755

31.1 Introduction755

31.2 TotalReturnSwaps759

31.3 CreditSpreadOptions/Forwards763

31.4 CreditDefaultSwaps763

31.5 Credit-LinkedNotes772

31.6 CorrelationProducts775

31.7 Summary781

31.8 Exercises782

Appendix31A TheCDSBigBang784

Chapter32

StructuralModelsofDefaultRisk789

32.1 Introduction789

32.2 TheMerton(1974)Model790

32.3 IssuesinImplementation799

32.4 APractitionerModel804

32.5 ExtensionsoftheMertonModel806

32.6 EvaluationoftheStructural ModelApproach808

32.7 Summary810

32.8 Exercises811

Appendix32A TheDelianedis-Geske Model813

Chapter33

Reduced-FormModelsofDefaultRisk816

33.1 Introduction816

33.2 ModelingDefaultI:IntensityProcesses817

33.3 ModelingDefaultII:RecoveryRate Conventions821

33.4 TheLitterman-IbenModel823

33.5 TheDuffie-SingletonResult828

33.6 DefaultableHJMModels830

33.7 Ratings-BasedModeling:TheJLT Model832

33.8 AnApplicationofReduced-FormModels: PricingCDS840

33.9 Summary842

33.10 Exercises842

Appendix33A Duffie-Singleton inDiscreteTime846

Appendix33B DerivationoftheDrift-Volatility Relationship847

Chapter34

ModelingCorrelatedDefault850

34.1 Introduction850

34.2 ExamplesofCorrelatedDefault Products850

34.3 SimpleCorrelatedDefaultMath852

34.4 StructuralModelsBasedon AssetValues855

34.5 Reduced-FormModels861

34.6 MultiperiodCorrelatedDefault862

34.7 FastComputationofCreditPortfolioLoss DistributionswithoutSimulation865

34.8 CopulaFunctions868

34.9 Top-DownModelingofCredit PortfolioLoss880

34.10 Summary884

34.11 Exercises885

BibliographyB-1 IndexI-1

ThefollowingWebchaptersare availableatwww.mhhe.com/sd2e:

PARTSIX

Computation1

Chapter35

DerivativePricingwithFinite Differencing3

35.1 Introduction3

35.2 SolvingDifferentialEquations4

35.3 AFirstApproachtoPricingEquity Options7

35.4 ImplicitFiniteDifferencing13

35.5 TheCrank-NicholsonScheme17

35.6 FiniteDifferencingforTerm-Structure Models19

35.7 Summary21

35.8 Exercises22

Chapter36

DerivativePricingwithMonteCarlo Simulation23

36.1 Introduction23

36.2 SimulatingNormalRandomVariables24

36.3 BivariateRandomVariables25

36.4 CholeskyDecomposition25

36.5 StochasticProcessesforEquityPrices27

36.6 ARCHModels29

36.7 Interest-RateProcesses30

36.8 EstimatingHistoricalVolatilityfor Equities32

36.9 EstimatingHistoricalVolatilityforInterest Rates32

36.10 Path-DependentOptions33

36.11 VarianceReduction35

36.12 MonteCarloforAmericanOptions38

36.13 Summary42

36.14 Exercises43

Chapter37 Using Octave 45

37.1 SomeSimpleCommands45

37.2 RegressionandIntegration48

37.3 ReadinginData,Sorting,andFinding50

37.4 EquationSolving55

37.5 Screenshots55

AuthorBiographies

RangarajanK.(“Raghu”)Sundaram isProfessorofFinanceatNewYorkUniversity’sSternSchoolofBusiness.Hewaspreviouslyamemberoftheeconomicsfaculty attheUniversityofRochester.Raghuhasanundergraduatedegreeineconomicsfrom LoyolaCollege,UniversityofMadras;anMBAfromtheIndianInstituteofManagement, Ahmedabad;andaMaster’sandPh.D.ineconomicsfromCornellUniversity.Hewascoeditorofthe JournalofDerivatives from2002–2008andisorhasbeenamemberofseveral othereditorialboards.Hisresearchinfinancecoversarangeofareasincludingagency problems,executivecompensation,derivativespricing,creditriskandcreditderivatives, andcorporatefinance.Hehasalsopublishedextensivelyinmathematicaleconomics,decisiontheory,andgametheory.Hisresearchhasappearedinallleadingacademicjournalsin financeandeconomictheory.TherecipientoftheJensenAwardandafinalistfortheBrattle Prizeforhisresearchinfinance,Raghuhasalsowonseveralteachingawardsincluding,in 2007,theinauguralDistinguishedTeachingAwardfromtheSternSchoolofBusiness.This isRaghu’ssecondbook;hisfirst,aPh.D.-leveltexttitled AFirstCourseinOptimization Theory,waspublishedbyCambridgeUniversityPress.

SanjivDas istheWilliamandJaniceTerryProfessorofFinanceatSantaClaraUniversity’s LeaveySchoolofBusiness.Hepreviouslyheldfacultyappointmentsasassociateprofessor atHarvardBusinessSchoolandUCBerkeley.Heholdspost-graduatedegreesinfinance (M.PhilandPh.D.fromNewYorkUniversity),computerscience(M.S.fromUCBerkeley), anMBAfromtheIndianInstituteofManagement,Ahmedabad,B.Cominaccountingand economics(UniversityofBombay,SydenhamCollege),andisalsoaqualifiedcostand worksaccountant.Heisasenioreditorof TheJournalofInvestmentManagement,coeditorof TheJournalofDerivatives andthe JournalofFinancialServicesResearch,and associateeditorofotheracademicjournals.Heworkedinthederivativesbusinessinthe Asia-Pacificregionasavice-presidentatCitibank.Hiscurrentresearchinterestsinclude themodelingofdefaultrisk,machinelearning,socialnetworks,derivativespricingmodels, portfoliotheory,andventurecapital.Hehaspublishedovereightyarticlesinacademic journals,andhaswonnumerousawardsforresearchandteaching.Hecurrentlyalsoserves asaseniorfellowattheFDICCenterforFinancialResearch.

Preface

Thetwoofushaveworkedtogetheracademicallyformorethanaquartercentury,firstas graduatestudents,andthenasuniversityfaculty.Givenourclosecollaboration,ourcommon researchandteachinginterestsinthefieldofderivatives,andthefrequentpedagogical discussionswehavehadonthesubject,thisbookwasperhapsinevitable.

Thefinalproductgrewoutofmanysources.Aboutthree-fourthsofthebookwasdevelopedbyRaghufromhisnotesforhisderivativescourseatNewYorkUniversityaswellas forotheracademiccoursesandprofessionaltrainingprogramsatCreditSuisse,ICICIBank, theInternationalMonetaryFund(IMF),Invesco-GreatWall,J.P.Morgan,MerrillLynch, theIndianSchoolofBusiness(ISB),theInstituteforFinancialManagementandResearch (IFMR),andNewYorkUniversity,amongotherinstitutions.Otherpartsweredeveloped byacademiccoursesandprofessionaltrainingprogramstaughtbySanjivatHarvardUniversity,SantaClaraUniversity,theUniversityofCaliforniaatBerkeley,theISB,theIFMR, theIMF,andCitibank,amongothers.Somechaptersweredevelopedspecificallyforthis book,asweremostoftheend-of-chapterexercises.

Thediscussionbelowprovidesanoverviewofthebook,emphasizingsomeofitsspecial features.Weprovidetoooursuggestionsforvariousderivativescoursesthatmaybecarved outofthebook.

AnOverviewoftheContents

Themainbodyofthisbookisdividedintosixparts.Parts1–3cover,respectively,futuresand forwards;options;andswaps.Part4examinesterm-structuremodelingandthepricingof interest-ratederivatives,whilePart5isconcernedwithcreditderivativesandthemodeling ofcreditrisk.Part6discussescomputationalissues.Adetaileddescriptionofthebook’scontentsisprovidedinSection1.5;here,weconfineourselvestoabriefoverviewofeachpart.

Part1 examinesforwardandfuturescontracts,Thetopicscoveredinthisspaninclude thestructureandcharacteristicsoffuturesmarkets;thepricingofforwardsandfutures; hedgingwithforwardsandfutures,inparticular,thenotionof minimum-variancehedging anditsimplementation;andinterest-rate-dependentforwardsandfutures,suchasforwardrateagreementsorFRAs,eurodollarfutures,andTreasuryfuturescontracts.

Part2,thelengthiestportionofthebook,isconcernedmainlywithoptions.Webegin withadiscussionofoptionpayoffs,theroleofvolatility,andtheuseofoptionsinincorporatingintoaportfoliospecificviewsonmarketdirectionand/orvolatility.Thenweturn ourattentiontothepricingofoptionscontracts.ThebinomialandBlack-Scholesmodels aredevelopedindetail,andseveralgeneralizationsofthesemodelsareexamined.From pricing,wemovetohedgingandadiscussionoftheoption“greeks,”measuresofoption sensitivitytochangesinthemarketenvironment.Roundingoffthepricingandhedging material,twochaptersdiscussawiderangeof“exotic”optionsandtheirbehavior.

TheremainderofPart2focusesonspecialtopics:portfoliomeasuresofrisksuchas Value-at-Riskandthenotionofriskbudgeting,thepricingandhedgingofconvertiblebonds, andastudyof“real”options,optionalitiesembeddedwithininvestmentprojects.

Part3 ofthebooklooksatswaps.Theusesandpricingofinterestrateswapsare coveredindetail,asareequityswaps,currencyswaps,andcommodityswaps.(Otherinstrumentsbearingthe“swaps”monikerarecoveredelsewhereinthebook.Varianceand volatilityswapsarepresentedinthechapteronBlack-Scholes,andcredit-defaultswapsand

total-returnswapsareexaminedinthechapteroncredit-derivativeproducts.)Alsoincluded inPart3isapresentationofcaps,floors,andswaptions,andofthe“marketmodel”usedto pricetheseinstruments.

Part4 dealswithinterest-ratemodeling.Webeginwithdifferentnotionsoftheyield curve,theestimationoftheyieldcurvefrommarketdata,andthechallengesinvolvedin modelingmovementsintheyieldcurve.Wethenworkourwaythroughfactormodelsof theyieldcurve,includingseveralwell-knownmodelssuchasHo-Lee,Black-Derman-Toy, Vasicek,Cox-Ingersoll-Ross,andothers.AfinalchapterpresentstheHeath-Jarrow-Morton framework,andalsothatoftheLiborandswapmarketmodels.

Part5 dealswithcreditriskandcreditderivatives.Anopeningchapterprovidesa taxonomyofproductsandtheircharacteristics.Theremainingchaptersareconcernedwith modelingcreditrisk.Structuralmodelsarecoveredinonechapter,reduced-formmodels inthenext,andcorrelated-defaultmodelinginthethird.

Part6,availableonlineat http://www.mhhe.com/sd1e,looksatcomputationalissues. Finite-differencingandMonteCarlomethodsarediscussedhere.Afinalchapterprovides atutorialontheuseof Octave,afreesoftwareprogramakinto Matlab,thatweusefor illustrativepurposesthroughoutthebook.

BackgroundKnowledge

Itwouldbeinaccuratetosaythatthisbookdoesnotpresupposeanyknowledgeonthe partofthereader,butitistruethatitdoesnotpresupposemuch.Abasicknowledgeof financialmarkets,instruments,andvariables(equities,bonds,interestrates,exchangerates, etc.)willobviouslyhelp—indeed,isalmostessential.Sotoowilladegreeofanalytical preparedness(forexample,familiaritywithlogsandexponents,compounding,present valuecomputations,basicstatisticsandprobability,thenormaldistribution,andsoon).But beyondthis,notmuchisrequired.Thebookislargelyself-contained.Theuseofadvanced (fromthestandpointofanMBAcourse)mathematicaltools,suchasstochasticcalculus,is kepttoaminimum,andwheresuchconceptsareintroduced,theyareoftendeviationsfrom themainnarrativethatmaybeavoidedifsodesired.

WhatIsDifferentaboutThisBook?

Ithasbeenourexperiencethattheoverwhelmingmajorityofstudentsinderivativescourses goontobecometraders,creatorsofstructuredproducts,orotherusersofderivatives,for whomadeepconceptual,ratherthansolelymathematical,understandingofproductsand modelsisrequired.Happily,thefieldofderivativeslendsitselftosuchanend:while itisoneofthemostmathematicallysophisticatedareasoffinance,itisalsopossible, perhapsmoresothaninanyotherareaoffinance,toexplainthefundamentalprinciples underlyingderivativespricingandrisk-managementinsimple-to-understandandrelatively non-mathematicalterms.Ourbooklookstocreatepreciselysuchablendedapproach,one thatisformalandrigorous,yetintuitiveandaccessible.

Tothispurpose,agreatdealofoureffortthroughoutthisbookisspentonexplaining whatliesbehindtheformalmathematicsofpricingandhedging.Howareforwardprices determined?WhydoestheBlack-Scholesformulahavetheformitdoes?Whatistheoption gammaandwhyisitofsuchimportancetoatrader?Theoptiontheta?Whydoterm-structure modelstaketheapproachtheydo?Inparticular,whatarethesubtletiesandpitfallsin modelingterm-structuremovements?Howmayequitypricesbeusedtoextractdefaultrisk ofcompanies?Debtprices?Howdoesdefaultcorrelationmatterinthepricingofportfolio creditinstruments?Whydoesitmatterinthisway?Inallofthesecasesandothersthroughout

thebook,weuseverbalandpictorialexpositions,andsometimessimplemathematical models,toexplaintheunderlyingprinciplesbeforeproceedingtoaformalanalysis.

Noneofthisshouldbetakentoimplythatourpresentationsareinformalormathematicallyincomplete.Butitistruethatweeschewtheuseofunnecessarymathematics.Where discrete-timesettingscanconveythebehaviorofamodelbetterthancontinuous-timesettings,weresorttosuchaframework.Whereapicturecandotheworkofathousand(oreven ahundred)words,weuseapicture.Andweavoidthepresentationof“blackbox”formulae tothemaximumextentpossible.Inthefewcaseswherederivingthepricesofsomederivativeswouldrequiretheuseofadvancedmathematics,wespendeffortexplainingintuitively theformandbehaviorofthepricingformula.

Tosupplementtheintuitiveandformalpresentations,wemakeextensiveuseofnumerical examplesforillustrativepurposes.Toenablecomparability,thenumericalexamplesare oftenbuiltaroundacommonparametrization.Forexample,inthechapteronoptiongreeks, abaselinesetofparametervaluesischosen,andthebehaviorofeachgreekisillustrated usingdeparturesfromthesebaselines.

Inaddition,thebookpresentsseveralfull-lengthcasestudies,includingsomeofthemost (in)famousderivativesdisastersinhistory.TheseincludeAmaranth,Barings,Long-Term CapitalManagement(LTCM),Metallgesellschaft,Procter&Gamble,andothers.These aresupplementedbyothercasestudiesavailableonthisbook’swebsite,includingAshanti, Sumitomo,theSon-of-Bosstaxshelters,andAmericanInternationalGroup(AIG).

Finally,sincethebestwaytolearnthetheoryofderivativespricingandhedgingisby workingthroughexercises,thebookoffersalargenumberofend-of-chapterproblems. Theseproblemsareofthreetypes.Someareconceptual,mostlyaimedatensuringthebasic definitionshavebeenunderstood,butoccasionallyalsoinvolvingalgebraicmanipulations. Thesecondgroupcomprisenumericalexercises,problemsthatcanbesolvedwithacalculatororaspreadsheet.Thelastgroupareprogrammingquestions,questionsthatchallenge thestudentstowritecodetoimplementspecificmodels.

NewtothisEdition

Thiseditionhasbeensubstantiallyrevisedandincorporatesmanyadditionstoandchanges fromtheearlierone,entirelycarriedoutbythefirstauthor.Theseinclude

•brieftolengthydiscussionsofseveralnewcasestudies(e.g.,AracruzCellulose’s$1 billion + lossesfromforeign-exchangederivativesin2008,Soci´et´eG´en´erale’s €5billion lossesfromJ´erômeKerviel’s“unauthorized”derivativestradingin2008,HarvardUniversity’s$1.25billionlossesfromswapcontractsin2009–13,thelikelystructureofthe GoldmanSachs-Greeceswaptransactionof2002thatallowedGreecetocircumventEU restrictionsondebt,andothers);

•expandedexpositionsofseveralkeytheoreticalconcepts(suchtheBlack-ScholesformulainChapter14);

•detaileddiscussionsofchangingmarketpractices(suchasthenew“dualcurve”approach toswappricinginChapter23andthecredit-eventauctionsthatarehardwiredintoall credit-defaultswapcontractspost-2009inChapter31);

•newdescriptionsofexchange-tradedinstrumentsandindices(e.g.,theCBoT’sUltra T-BondfuturesinChapter6,theCBOE’sBXMandBXY“coveredcall”indicesin Chapter8ortheCBOE’sS&P500andVIXdigitaloptionsinChapter18);

•and,ofcourse,thankstotheassistanceofstudentsandcolleagues,theidentificationand correctionoftypographicalerrors.

Specialthankstoallthosewhosentintheircommentsandsuggestionsonthefirstedition. Wetrusttheend-productismoresatisfying.

PossibleCourseOutlines

Figure1describesthelogicalflowofchaptersinthebook.Thebookcanbeusedatthe undergraduateandMBAlevelsasthetextforafirstcourseinderivatives;forasecond(or advanced)courseinderivatives;fora“topics”courseinderivatives(asafollow-uptoafirst course);andforafixed-incomeand/orcreditderivativescourse;amongothers.Wedescribe belowoursuggestedselectionofchaptersforeachofthese.

Afirstcourseinderivativestypicallycoversforwardsandfutures,basicoptionsmaterial, andperhapsinterestrateswaps.SuchacoursecouldbebuiltaroundChapters1–4onfutures marketsandforwardandfuturespricing;Chapters7–14onoptionspayoffsandtrading strategies,no-arbitragerestrictionsandput-callparity,andthebinomialandBlack-Scholes models;Chapters17–19onoptiongreeksandexoticoptions;andChapter23oninterest rateswapsandotherfloating-rateproducts.

Asecondcourse,focusedprimarilyoninterest-rateandcredit-riskmodeling,couldbegin withareviewofbasicoptionpricing(Chapters11–14),moveontoanexaminationofmore complexpricingmodels(Chapter16),thencoverinterest-ratemodeling(Chapters26–30) andfinallycreditderivativesandcredit-riskmodeling(Chapters31–34).

A“topics”coursefollowingthefirstcoursecouldbeginagainwithareviewofbasicoptionpricing(Chapters11–14)followedbyanexaminationofmorecomplexpricingmodels (Chapter16).ThiscouldbefollowedbyValue-at-Riskandrisk-budgeting(Chapter20); convertiblebonds(Chapter21);realoptions(Chapter22);andinterest-rate,equity,and currencyswaps(Chapters23–25),withthefinalpartofthecoursecoveringeitheranintroductiontoterm-structuremodeling(Chapters26–28)oranintroductiontocreditderivatives andstructuralmodels(Chapters31and32).

Finally,acourseonfixed-incomederivativescanbestructuredaroundbasicforward pricing(Chapter3);interest-ratefuturesandforwards(Chapter6);basicoptionpricingand theBlack-Scholesmodel(Chapters11and14);interestrateswaps,caps,floors,andswaptions,andtheBlackmodel(Chapter23);andtheyieldcurveandterm-structuremodeling (Chapters26–30).

AFinalComment

Thisbookhasbeenseveralyearsinthemakingandhasundergoneseveralrevisionsinthat time.Meanwhile,thederivativesmarkethasitselfbeenchangingatanexplosivepace.The financialcrisisthateruptedin2008willalmostsurelyresultinalteringmajorcomponents ofthederivativesmarket,particularlyinthecaseofover-the-counterderivatives.Thus,itis possiblethatsomeoftheproductswehavedescribedcouldvanishfromthemarketinafew years,orthewaytheseproductsaretradedcouldfundamentallychange.Butthe principles governingthevaluationandrisk-managementoftheseproductsaremorepermanent,and itisthoseprinciples,ratherthansolelythedetailsoftheproductsthemselves,thatwehave triedtocommunicateinthisbook.Wehaveenjoyedwritingthisbook.Wehopethereader findsthefinalproductasenjoyable.

Acknowledgments

Wehavebenefitedgreatlyfrominteractionswithanumberofourcolleaguesinacademia andothersinthebroaderfinanceprofession.Itisapleasuretobeabletothankthemin print.

AtNewYorkUniversity,whereRaghucurrentlyteachesandSanjivdidhisPhD(and hasbeenafrequentvisitorsince),wehaveenjoyedmanyilluminatingconversationsover theyearsconcerningderivativesresearchandteaching.Forthese,wethankViralAcharya, EdAltman,YakovAmihud,MenachemBrenner,AswathDamodaran,SteveFiglewski, HalinaFrydman,KoseJohn,TonySaunders,andMartiSubrahmanyam.Weowespecial thankstoViralAcharya,long-timecollaboratorofbothauthors,forhisfeedbackonearlier versionsofthisbook;EdAltman,fromwhomwe—liketherestoftheworld—learneda greatdealaboutcreditriskandcreditmarkets,andwhowasalwaysgenerouswithhistime andsupport;MenachemBrenner,formanydelightfulexchangesconcerningderivatives usageandstructuredproducts;SteveFiglewski,withwhomwewereprivilegedtoserveas co-editorsofthe JournalofDerivatives,awonderfullearningexperience;and,especially, MartiSubrahmanyam,whowasSanjiv’sPhDadvisoratNYUandwithwhomRaghuhas co-taughtexecutive-MBAandPhDcoursesonderivativesandcreditriskatNYUsince themid-90s.Marti’semphasisonanintuitiveunderstandingofmathematicalmodelshas considerablyinfluencedbothauthors’approachtotheteachingofderivatives;itseffectmay beseenthroughoutthisbook.

AtSantaClaraUniversity,GeorgeChacko,AtulyaSarin,HershShefrin,andMeir Statmanallprovidedmuch-appreciatedadvice,support,andencouragement.Valuableinput alsocamefromothersintheacademicprofession,includingMarcoAvellaneda,Pierluigi Balduzzi,JonathanBerk,DarrellDuffie,AnuragGupta,PaulHanouna,NikunjKapadia, DanOstrov,N.R.Prabhala,andRamanUppal.Inthebroaderfinancecommunity,wehave benefitedgreatlyfrominteractionswithSanthoshBandreddi,JamilBaz,RichardCantor, GiffordFong,SilverioForesi,GaryGeng,GraceKoo,ApoorvaKoticha,MuraliKrishna, MarcoNaldi,ShankarNarayan,RajRajaratnam,RahulRathi,JacobSisk,RogerStein, andRamSundaram.ThefirstauthorwouldparticularlyliketothankRamSundaramand MuraliKrishnafornumerousstimulatingandinformativeconversationsconcerningthe markets;thesecondauthorthanksRobertMertonforhisinsightsonderivativesandguidanceinteachingcontinuous-timefinance,andGiffordFongformanyyearsofgenerous mentorship.

Overtheyearsthatthisbookwasbeingwritten,manyofourcolleaguesintheprofessionprovided(anonymous)reviewsthatgreatlyhelpedshapethefinalproduct.Avery specialthankstothosereviewerswhotookthetimetoreviewvirtuallyeverychapterindraft form:BalaArshanapalli(IndianaUniversity–Northwest),Dr.R.BrianBalyeat(TexasA&M University),JamesBennett(UniversityofMassachusetts–Boston),Jinliang(Jack)Li(NortheasternUniversity),SpencerMartin(ArizonaStateUniversity),PatriciaMatthews(Mount UnionCollege),DennisOzenbas(MontclairStateUniversity),VivekPandey(University ofTexas–Tyler),PeterRitchken(Case-WesternReserveUniversity),TieSu(University ofMiami),ThomasTallerico(DowlingCollege),KudretTopyan(ManhattanCollege), AlanTucker(PaceUniversity),JorgeUrrutia(LoyolaUniversity–Watertower),MattWill (UniversityofIndianapolis),andGuofuZhou(WashingtonUniversity–St.Louis).

Aswehavenotedinthepreface,thisbookgrewoutofnotesdevelopedbytheauthorsfor academiccoursesandprofessionaltrainingprogramsatanumberofinstitutionsincluding

HarvardUniversity,SantaClaraUniversity,UniversityofCaliforniaatBerkeley,Citibank, Credit-Suisse,MerrillLynch,theIMF,and,mostofall,NewYorkUniversity.Participants inallofthesecourses(andatLondonBusinessSchool,whereanearlierversionofRaghu’s NYUnoteswereusedbyViralAcharya)haveprovideddetailedfeedbackthatledtoseveral revisionsoftheoriginalmaterial.Wegreatlyappreciatethecontributiontheyhavemadeto thefinalproduct.WearealsogratefultoRaviKumarofCapitalMetricsandRiskSolutions (P)Ltd.forhisterrificassistanceincreatingthesoftwarethataccompaniesthisbook;andto PriyankaSinghofthesameorganizationforproofreadingthemanuscriptanditsexercises.

AspecialthankstotheteamatMcGraw(especiallyLoriBradshaw,ChuckSynovec, JenniferUpton,andMaryJaneLampe)forthesplendidsupportwereceived.Thankstooto SusanNortonforhermeticulouscopyeditingjob;AmyHillforhercarefulproofreading; andMohammadMisbahforthepatienceandcarewithwhichheguidedthisbookthrough thetypesettingprocess.

Ourgreatestdebtsaretothemembersofourrespectivefamilies.Wearebothextraordinarilyfortunateinhavinglargeandsupportiveextendedfamilynetworks.Toallofthem: thankyou.Weoweyoumorethanwecaneverrepay.

RangarajanK.Sundaram

Chapter 1 Introduction

Theworldderivativesmarketisan immense one.TheBankforInternationalSettlements (BIS)estimatedthatinJune2012,thetotalnotionaloutstandingamountworldwidewasa staggering$639 trillion withacombinedmarketvalueofover$25trillion(Table1.1)— andthisfigureincludesonlyover-the-counter(OTC)derivatives,thosederivativestraded directlybetweentwoparties.Itdoesnotcountthetrillionsofdollarsinderivativesthatare tradeddailyontheworld’smanyexchanges.Bywayofcomparison,worldGDPin2011 wasestimatedatjustunder$70trillion.

Formuchofthelasttwodecades,growthhasbeenfurious.Totalnotionaloutstanding inOTCderivativesmarketsworldwideincreasedalmost tenfold inthedecadefrom1998 to2008(Table1.2).Derivativesturnoverontheworld’sexchangesquadrupledbetween 2001and2007,reachingavolumeofover$2.25 quadrillion inthelastyearofthatspan (Table1.3).Marketsfellwiththeonsetofthefinancialcrisis,butby2011–12,asubstantial portionofthatdeclinehadbeenreversed.

Thegrowthhasbeentrulywidespread.Therearenowthrivingderivativesexchangesnot onlyinthetraditionaldevelopedeconomiesofNorthAmerica,Europe,andJapan,butalso inBrazil,China,India,Israel,Korea,Mexico,andSingapore,amongmanyothercountries. AsurveybytheInternationalSwapsandDerivativesAssociation(ISDA)in2003found that92%oftheworld’s500largestcompaniesusederivativestomanageriskofvarious forms,especiallyinterest-raterisk(92%)andcurrencyrisk(85%),but,toalesserextent, alsocommodityrisk(25%)andequityrisk(12%).Firmsinover90%ofthecountries representedinthesampleusedderivatives.

Matching—andfueling—thegrowthhasbeenthepaceofinnovationinthemarket. Traditionalderivativeswerewrittenoncommodityprices,butbeginningwithforeigncurrencyandotherfinancialderivativesinthe1970s,newformsofderivativeshavebeenintroducedalmostcontinuously.Today,derivativescontractsreferenceawiderangeofunderlying instrumentsincludingequityprices,commodityprices,exchangerates,interestrates,bond prices,indexlevels,andcreditrisk.Derivativeshavealsobeenintroduced,withvaryingsuccessrates,onmoreexoticunderlyingvariablessuchasmarketvolatility,electricityprices, temperaturelevels,broadband,newsprint,andnaturalcatastrophes,amongmanyothers.

Thisisanimpressivepicture.Onceasideshowinworldfinancialmarkets,derivatives havetodaybecomekeyinstrumentsofrisk-managementandpricediscovery.Yetderivatives havealsobeenthetargetoffiercecriticism.In2003,WarrenBuffet,perhapstheworld’smost successfulinvestor,labeledthem“financialweaponsofmassdestruction.”Derivatives— especiallycreditderivatives—havebeenwidelyblamedforenabling,oratleastexacerbating, theglobalfinancialmarketscrisisthatbeganinlate2007.Victimsofderivatives(mis-)use overthedecadesincludesuchprominentnamesasthecenturies-oldBritishmerchantbank Barings,theGermanindustrialconglomerateMetallgesellschaftAG,theJapanesetrading

TABLE1.1 BISEstimatesofOTCDerivativesMarketsNotionalOutstandingandMarketValues:2008–12 (FiguresinUSDbillions)

Source:BISwebsite(http://www.bis.org).

powerhouseSumitomo,thegiantUSinsurancecompany,AmericanInternationalGroup (AIG),andBrazil’sAracruz,thentheworld’slargestmanufacturerofeucalyptuspulp. What is aderivative?Whatarethedifferenttypesofderivatives?Whatarethebenefits ofderivativesthathavefueledtheirgrowth?Therisksthathaveledtodisasters?Howis thevalueofaderivativedetermined?Howaretherisksinaderivativemeasured?How cantheserisksbemanaged(or hedged)?Theseandotherquestionsarethefocusofthis book.Wedescribeandanalyzeawiderangeofderivativesecurities.Bycombiningthe analyticaldescriptionswithnumericalexamples,exercises,andcasestudies,wepresentan introductiontotheworldofderivativesthatisatonceformalandrigorousyetaccessible andintuitive.Therestofthischapterelaboratesandlaysthefoundationforthebook.

WhatAreDerivatives?

A derivativesecurity isafinancialsecuritywhosepayoffdependson(or derivesfrom)other, morefundamental,variablessuchasastockprice,anexchangerate,acommodityprice, aninterestrate—oreventhepriceofanotherderivativesecurity.Theunderlyingdriving variableiscommonlyreferredtoassimply theunderlying. Thesimplestkindofderivative—andhistoricallytheoldestform,datingbackthousands ofyears—isa forwardcontract.Aforwardcontractisoneinwhichtwoparties(commonly referredtoasthe counterparties inthetransaction)agreetothetermsofatradetobe consummatedonaspecifieddateinthefuture.Forexample,onDecember3,abuyerand sellermayenterintoaforwardcontracttotradein100ozofgoldinthreemonths(i.e.,on March3)atapriceof$1,500/oz.Inthiscase,thesellerisundertakingtosell100ozin threemonthsatapriceof$1,500/ozwhilethebuyerisundertakingtobuy100ozofgold inthreemonthsat$1,500/oz.

TABLE1.2 BISEstimatesofOTCDerivativesMarketsNotionalOutstanding:1998–2012 (FiguresinUSDbillions)

Jun1998Jun2002Jun2006Jun2008Jun2010Jun2011Jun2012

TotalContracts 72,134127,509372,513672,558582,685706,884638,928

FXDerivatives 18,71918,06838,12762,98353,15364,69866,645 ForwardsandFXswaps12,14910,42619,40731,96625,62431,11331,395 Currencyswaps 1,9474,2159,69616,30716,36022,22824,156 Options 4,6233,4279,02414,71011,17011,35811,094 Interest-RateDerivatives42,36889,955262,526458,304451,831553,240494,018 Forwardrateagreements5,1479,14618,11739,37056,24255,74764,302 Interestrateswaps 29,36368,234207,588356,772347,508441,201379,401 Options 7,85812,57536,82162,16248,08156,29150,314 EquityDerivatives 1,2742,2146,78210,1776,2606,8416,313 Forwardsandswaps 1543861,4302,6571,7542,0291,880 Options 1,1201,8285,3517,5214,5064,8134,434 CommodityDerivatives4437776,39413,2292,8523,1972,993 Gold 185279456649417468523 Othercommodities 2584985,93812,5802,4342,7292,470 Forwardsandswaps

Source:BISwebsite(http://www.bis.org).

Acommonmotivationforenteringintoaforwardcontractistheeliminationofcash-flow uncertaintyfromafuturetransaction.Inourexample,ifthebuyeranticipatesaneedfor 100ozofgoldinthreemonthsandisworriedaboutpricefluctuationsoverthatperiod, anyuncertaintyaboutthecashoutlayrequiredcanberemovedbyenteringintoaforward contract.Similarly,ifthesellerexpectstobeoffloading100ozofgoldinthreemonths andisconcernedaboutpricesthatmightprevailattheendofthathorizon,enteringintoa forwardcontractlocksinthepricereceivedforthatfuturesale.

Inshort,forwardcontractsmaybeusedasinstrumentsof hedging.Infinancialparlance, hedgingisthereductionincash-flowriskassociatedwithamarketcommitment.Forward contractsarecommonlyusedbyimportersandexportersworriedaboutexchange-rate fluctuations,investorsandborrowersworriedaboutinterest-ratefluctuations,commodity producersandbuyersworriedaboutcommoditypricefluctuations,andsoon.

Aslightlymorecomplexexampleofaderivativeisan option.Asinaforward,anoption contracttoospecifiesthetermsofafuturetrade,butwhileaforwardcommitsbothpartiesto thetrade,inanoption,onepartytothecontract(calledthe holder oftheoption)retainsthe right toenforceoroptoutofthecontract.Iftheholderhastherightto buy atthespecified price,theoptioniscalleda calloption;iftherightto sell atthatprice,a putoption

Thekeydifferencebetweenaforwardandanoptionisthatwhileaforwardcontractis aninstrumentfor hedging,anoptionprovidesaformoffinancial insurance.Consider,for example,acalloptionongoldinwhichthebuyerhastherighttobuygoldfromthesellerata priceof(say)$1,500/ozinthreemonths’time.Ifthepriceofgoldinthreemonthsisgreater than$1,500/oz(forexample,itis$1,530/oz),thenthebuyerwillexercisetherightinthecontractandbuythegoldforthecontractpriceof$1,500.However,ifthepriceinthreemonths

islessthan$1,500/oz(e.g.,is$1,480/oz),thebuyercanchoosetooptoutofthecontract and,ifnecessary,buythegolddirectlyinthemarketatthecheaperpriceof$1,480/oz.

Thus,holdingacalloptioneffectivelyprovidesthebuyerwithprotection(or“insurance”) againstan increase inthepriceabovethatspecifiedinthecontractevenwhileallowingthe buyertotakefulladvantageofpricedecreases.Sinceitisthesellerwhotakestheotherside ofthecontractwheneverthebuyerdecidestoenforceit,itisthesellerwhoprovidesthis insurancetothebuyer.Inexchangeforprovidingthisprotection,thesellerwillchargethe buyeranup-frontfeecalledthe calloptionpremium.

Analogously,a put optionprovidesthesellerwithinsuranceagainsta decrease inthe price.Forinstance,consideraputoptionongoldinwhichthesellerhastherighttosellgold tothebuyerat$1,500/oz.Ifthepriceofgoldfallsbelow$1,500/oz,thesellercanexercise therightintheputandsellthegoldfor$1,500/oz,butifthepriceofgoldrisestomore than$1,500/oz,thenthesellercanelecttolettheputlapseandsellthegoldatthehigher marketprice.Holdingtheputinsurestheselleragainstafallinthepricebelow$1,500/oz. Thebuyerprovidesthisinsuranceandwillchargeanup-frontfee,the putpremium,for providingthisservice.

Optionsofferanalternativetoforwardsforinvestorsconcernedaboutfuturepricefluctuations.Unlikeforwards,thereisanup-frontcostofbuyinganoption(viz.,theoption premium)but,compensatingforthis,thereisnocompulsiontoexerciseifdoingsowould resultinaloss.

Forwardsandoptionsaretwoofthemostcommonandimportantformsofderivatives. Inmanyways,theyarethebuildingblocksofthederivativeslandscape.Manyotherforms ofderivativesexist,somewhicharesimplevariantsofthesestructures,othersmuchmore complexor“exotic”(thefavoredterminthederivativesareafordescribingsomethingthat isnotrun-of-the-millor“plainvanilla”).Weelaborateonthislaterinthischapterandin therestofthebook.Butfirst,wepresentabriefdiscussiononthedifferentcriteriathatmay beusedtoclassifyderivatives.

ClassifyingDerivatives

Therearethreepopularwaystoclassifyderivatives:bytheunderlying(equity,interestrate, etc.),bythenatureoftheinstrument(forwards,futures,options,etc.),andbythenatureof themarket(over-the-counterversusexchange-traded).

Apopularwaytoclassifyderivativesistogroupthemaccordingtotheunderlying.For example,an equityderivative isonewhoseunderlyingisanequitypriceorstockindex level;a currency or FX (shortforforeign-exchange) derivative isonewhoseunderlyingis anexchangerate;andsoon.Muchoftheworld’sderivativestradeonjustafewcommon underlyings.Table1.1showsthat interest-ratederivatives (derivativesdefinedoninterest ratesoroninterest-rate-sensitivesecuritiessuchasbonds)accountforaround75%ofthe grossmarketvalueoftheOTCderivativesmarket,withsmallersharesbeingtakenby currency,equity,commodity,andcreditderivatives.

Whilethesearethemostcommonunderlyings,derivativesmay,inprinciple,bedefined onjustaboutanyunderlyingvariable.Indeed,asubstantialchunkofthegrowthinderivatives marketsintheearlyyearsofthe2000scamefrom creditderivatives (derivativesdependent onthecreditriskofspecifiedunderlyingentities),acategoryofderivativesthatdidnot evenexistin1990.Asnotedearlierinthischapter,derivativeshavealsobeenintroduced onanumberofexoticunderlyingvariablesincludingelectricityprices,temperaturelevels, broadband,newsprint,andmarketvolatility.

Asecondpopularwaytoclassifyderivativesisusingthenatureofinstrument.Derivatives candiffergreatlyinthemannerinwhichtheydependontheunderlying,rangingfromvery simpledependenciestoverycomplexones.Nonetheless,mostderivativesfallintooneof twoclasses:thosethatinvolvea commitment toagiventradeorexchangeofcashflowsin

TABLE1.3 BISEstimatesofDerivativesTurnoveronExchanges:2001toQ3–2012 (FiguresinUSDbillions)

20012004200720092011ToQ3–2012

FuturesTotal445,683830,5461,584,6111,126,5191,524,141898,253

Interestrate420,950783,1401,433,7671,016,3621,359,131799,579 Currency 3,1587,40422,44224,59937,62824,576

Equityindex21,57540,001128,40185,559127,38274,098

NorthAmerica243,022438,718868,784599,025822,958487,000 Europe 154,094336,593588,087449,390565,185318,091 AsiaandPacific43,39448,547107,90763,125107,49072,875 Othermarkets5,1736,68719,83314,97928,50820,286

OptionsTotal148,370312,080701,891533,635635,363331,033 Interestrate122,766260,056547,629434,601466,281254,255

Currency 3565892,1411,9802,5251,776

Equityindex25,24751,435152,12197,054166,55775,002

NorthAmerica107,684181,503414,728216,390262,515158,288

Europe 33,473101,954203,787258,557258,265125,692

AsiaandPacific6,53427,57477,83652,751102,91239,708

Othermarkets6791,0485,5405,93611,6717,345

Source:BISwebsite(http://www.bis.org).

thefutureandthoseinwhichonepartyhasthe option toenforceoroptoutofthetradeor exchange.Includedintheformerclassarederivativesecuritiessuchas forwards, futures; and swaps;derivativesinthelatterclassarecalled options

Forwardsandoptionshavealreadybeendefinedabove.Futurescontractsaresimilar toforwardcontractsexceptthattheyaretradedonorganizedexchanges;wediscussthe differencesmorepreciselybelow.Swapsarecontractsinwhichthepartiescommitto multiple exchangesofcashflowsinthefuture,withthecashflowstobeexchangedcalculated underrulesspecifiedinthecontract;thus,swapsarelikeforwardsexceptwithmultiple transactionstowhichthepartiescommit.

Tables1.1–1.3usebothoftheseschemesofclassification,firstbreakingdowntheworld derivativesmarketbyunderlyingandthenintoforwards,futures,swaps,andoptions.The breakdownrevealssomeinterestingvariations.Forexample,whileswapsaccountforthe greatbulk(roughly80%)ofOTCinterest-ratederivatives,optionsconstituteover75%of OTCequityderivatives.

Athirdclassificationofderivativesisintoover-the-counter(OTC)orexchange-traded derivatives.Over-the-counterderivativescontractsaretradedbilaterallybetweentwocounterpartieswhodealdirectlywitheachother.Insuchtransactions,eachpartytakesthecredit riskoftheother(i.e.,theriskthattheothercounterpartymayfailtoperformor“default”on thecontract).Inexchange-tradedcontracts,thepartiesdealthoughanorganizedexchange, andtheidentityofthecounterpartyisusuallynotknown.Exchangescommonlyguarantee performanceonthecontract,soeachpartyistakingononlythecreditriskoftheexchange. ForwardsandswapsareOTCcontracts,whilefuturesareexchangetraded.Optionscanbe bothOTCandexchangetraded.

1.1ForwardandFuturesContracts

A forwardcontract isanagreementbetweentwopartiestotradeinaspecifiedquantityof aspecifiedgoodataspecifiedpriceonaspecifieddateinthefuture.Thefollowingbasic

terminologyisusedwhendiscussingthesecontracts:

•Thebuyerintheforwardcontractissaidtohavea longposition inthecontract;theseller issaidtohavea shortposition.

•Thegoodspecifiedinthecontractiscalledthe underlyingasset or,simply,the underlying

•Thedatespecifiedinthecontractonwhichthetradewilltakeplaceiscalledthe maturity date ofthecontract.

•Thepricespecifiedinthecontractforthetradeiscalledthe deliveryprice inthecontract. Thisisthepriceatwhichdeliverywillbemadebythesellerandacceptedbythebuyer.

Wewilldefinetheimportantconceptofa forwardprice shortly.Forthemoment,wenote thattheforwardpriceisrelatedto,butisnotthesameconceptas,thedeliveryprice.

Theunderlyinginaforwardcontractmaybeanycommodityorfinancialasset.Forward contractsmaybewrittenonforeigncurrencies,bonds,equities,orindices,orphysical commoditiessuchasoil,gold,orwheat.Forwardcontractsalsoexistonsuchunderlyings asinterestratesorvolatilitywhichcannotbedeliveredphysically(see,forexample,the forward-rateagreements orFRAsdescribedinChapter6,ortheforwardcontractsonmarket volatilityknownas variance and volatilityswaps, describedinChaper14);insuchcases, thecontractsaresettledincashwithonesidemakingapaymenttotheotherbasedonrules specifiedinthecontract.Cashsettlementisalsocommonlyusedforthoseunderlyingsfor whichphysicaldeliveryisdifficult,suchasequityindices.

Ashasbeendiscussed,aprimarymotiveforenteringintoaforwardcontractis hedging: usingaforwardcontractresultsinlocking-inapricetodayforafuturemarkettransaction, andthiseliminatescash-flowuncertaintyfromthetransaction.Foreigncurrencyforwards, forexample,enableexporterstoconvertthepaymentsreceivedinforeigncurrencyinto homecurrencyatafixedrate.Interest-rateforwardssuchasFRAsenablefirmstolock-in aninterestratetodayforafutureborrowingorinvestment.Commodityforwardssuchas forwardsonoilenableusersofoiltolock-inpricesatwhichfuturepurchasesaremadeand refinersofoiltolock-inapriceatwhichfuturesalesaremade.

Forwardcontractscanalsobeusedfor speculation,thatis,withoutanunderlyingexposurealreadyexisting.Aninvestorwhofeelsthatthepriceofsomeunderlyingislikelyto increasecanspeculateonthisviewbyenteringintoalongforwardcontractonthatunderlying.Ifpricesdogoupasanticipated,theinvestorcanbuytheassetatthelocked-inprice ontheforwardcontractandsellatthehigherprice,makingaprofit.Similarly,aninvestor wishingtospeculateonfallingpricescanuseashortforwardcontractforthispurpose.

KeyCharacteristicsofForwardContracts

Fourcharacteristicsofforwardcontractsdeservespecialemphasisbecausetheseareexactly thedimensionsalongwhichforwardsandfuturesdiffer:

•First,aforwardcontractisabilateralcontract.Thatis,thetermsofthecontractare negotiateddirectlybythesellerandthebuyer.

•Second,aforwardcontractiscustomizable.Thatis,thetermsofthecontract(maturity date,qualityoftheunderlying,etc.)canbe“tailored”totheneedsofthebuyerandseller.

•Third,thereispossibledefaultriskforbothparties.Eachpartytakestheriskthatthe othermayfailtoperformonthecontract.

•Fourth,neitherpartycantransferitsrightsandobligationsinthecontractunilaterallyto athirdparty.

Wereturntothesecharacteristicswhendiscussingfuturescontracts.