ToMichelle ToNilanjana
2.Futuresmarketsandcentralcounterparties...
3.Hedgingstrategiesusingfutures ......................................................................64 4.Interestrates. ...............................................................................................94
5.Determinationofforwardandfuturesprices.
23.Estimatingvolatilitiesandcorrelations.....
28.Martingalesandmeasures. ...........................................................................690
29.Interestratederivatives:Thestandardmarketmodels... ....................................708
30.Convexity,timing,andquantoadjustments... ..................................................728
31.Equilibriummodelsoftheshortrate .............................................................741
32.No-arbitragemodelsoftheshortrate...... ......................................................754
33.HJM,LMM,andmultiplezerocurves..... ......................................................777
34.SwapsRevisited.... ......................................................................................796
35.Energyandcommodityderivatives... .............................................................811
36.Realoptions. .............................................................................................829
37.Derivativesmishapsandwhatwecanlearnfromthem. ....................................843 Glossaryofterms. ......................................................................................857 DerivaGemsoftware..... ...............................................................................880
Majorexchangestradingfuturesandoptions ..................................................885 Tablesfor N ðxÞ ...........................................................................................886 Credits.. ....................................................................................................888 Authorindex. .............................................................................................889 Subjectindex.
ListofBusinessSnapshots
Chapter1.Introduction.
1.1Exchange-tradedmarkets..
1.2Over-the-countermarkets..
1.3Forwardcontracts.
1.4Futurescontracts..
1.5Options.
1.6Typesoftraders....
1.7Hedgers.
1.9Arbitrageurs..
1.10OverviewoftheIndianDerivativesMarket...
2.4Theoperationofmarginaccounts....
2.5OTCmarkets
2.6NewspaperQuotes
2.7Delivery
2.8Typesoftradersandtypesoforders.
2.9Regulation....
2.10Accountingandtax......
2.11Forwardvs.futurescontracts....
Chapter3.Hedgingstrategiesusingfutures.. ......................................................................64
3.1Basicprinciples... .................................................................................64
3.2Argumentsforandagainsthedging. ........................................................66
3.3Basisrisk.... ........................................................................................69
3.4Crosshedging..... .................................................................................73
3.5Stockindexfutures.. .............................................................................77
3.6Stackandroll......................................................................................84
Chapter4.Interestrates..
4.1Typesofrates.....
4.2Swaprates...........................................................................................99
4.3Therisk-freerate.
4.4Measuringinterestrates..
4.5Zerorates.........................................................................................104
4.6Bondpricing......
4.7Determiningzerorates....
4.9Forwardrateagreements.
4.10Duration.....
4.11Convexity....
4.12Theoriesofthetermstructureofinterestrates....
5.1Investmentassetsvs.consumptionassets....
5.2Shortselling
5.3Assumptionsandnotation......
5.4Forwardpriceforaninvestmentasset.
5.6Knownyield
5.7Valuingforwardcontracts......
5.8Areforwardpricesandfuturespricesequal?......
5.9Futurespricesofstockindices.
5.10Forwardandfuturescontractsoncurrencies......
5.11Futuresoncommodities..
5.12Thecostofcarry.
5.13Deliveryoptions..
5.14Futurespricesandexpectedfuturespotprices....
6.1Daycountandquotationconventions.
6.2Treasurybondfutures.....
6.3Eurodollarfutures. ..............................................................................167
6.4Duration-basedhedgingstrategiesusingfutures.... ...................................173
6.5Hedgingportfoliosofassetsandliabilities ..............................................174
Chapter7.Swaps.. .......................................................................................................180
7.1Mechanicsofinterestrateswaps...... .....................................................183
7.2Daycountissues... ..............................................................................188
7.3Confirmations...... ..............................................................................189
7.4Thecomparative-advantageargument...... ..............................................190
7.5Valuationofinterestrateswaps. ............................................................194
7.6Howthevaluechangesthroughtime .....................................................196
7.7Fixed-for-fixedcurrencyswaps.. ............................................................197
7.8Valuationoffixed-for-fixedcurrencyswaps... ..........................................200
7.9Othercurrencyswaps... .......................................................................202
7.10Creditrisk.... .....................................................................................203
7.11Creditdefaultswaps..... .......................................................................204
7.12Othertypesofswaps....
Chapter8.Securitizationandthecreditcrisisof2007... .....................................................212
8.1Securitization
8.2TheU.S.housingmarket..
Chapter9.XVAs..
9.1CVAandDVA...................................................................................227
Chapter10.Mechanicsofoptionsmarkets..
10.8Theoptionsclearingcorporation....
10.9Regulation.. ......................................................................................253
10.10Taxation..... ......................................................................................253
10.11Warrants,employeestockoptions,andconvertibles.... .............................254
10.12Over-the-counteroptionsmarkets...
Chapter11.Propertiesofstockoptions.. ...........................................................................261
11.1Factorsaffectingoptionprices.
11.3Upperandlowerboundsforoptionprices.. ...........................................266
11.4Put–callparity.... ...............................................................................268
11.5Callsonanon-dividend-payingstock.. ..................................................271
11.6Putsonanon-dividend-payingstock...
11.7Effectofdividends.. ...........................................................................276
Chapter12.Tradingstrategiesinvolvingoptions..... .............................................................282
12.1Principal-protectednotes. ....................................................................282
12.2Tradinganoptionandtheunderlyingasset
Chapter13.Binomialtrees .............................................................................................302
13.1Aone-stepbinomialmodelandano-arbitrageargument .........................302
13.2Risk-neutralvaluation..... ....................................................................306
13.3Two-stepbinomialtrees.. ....................................................................308
13.4Aputexample.... ...............................................................................311
13.5Americanoptions ...............................................................................312
13.6Delta... .............................................................................................313
13.7MatchingvolatilitywithUandD...
13.8Thebinomialtreeformulas.....
13.9Increasingthenumberofsteps .............................................................316
13.10UsingDerivaGem... ...........................................................................317
13.11Optionsonotherassets... ....................................................................318
Chapter14.WienerprocessesandIto ’slemma ...................................................................330
14.1TheMarkovproperty... .......................................................................330
14.2Continuous-timestochasticprocesses .....................................................331
14.3Theprocessforastockprice.... ............................................................336
14.4Theparameters..... ..............................................................................339
14.5Correlatedprocesses..... .......................................................................340
14.6Ito ’slemma. .....................................................................................341
14.7Thelognormalproperty .......................................................................342 Summary...... .....................................................................................343
Appendix:AnonrigorousderivationofIto ’slemma... ............................347
Chapter15.TheBlack–Scholes–Mertonmodel ...................................................................349
15.1Lognormalpropertyofstockprices.. .....................................................350
15.2Thedistributionoftherateofreturn .....................................................351
15.3Theexpectedreturn...... .......................................................................352
15.4Volatility...... .....................................................................................353
15.5TheideaunderlyingtheBlack–Scholes–Mertondifferentialequation ..........357
15.6DerivationoftheBlack–Scholes–Mertondifferentialequation.. .................359
15.7Risk-neutralvaluation.. .......................................................................362
15.8Black–Scholes–Mertonpricingformulas... ..............................................363
15.9Cumulativenormaldistributionfunction.. ..............................................366
15.10Warrantsandemployeestockoptions...... ..............................................367
15.11Impliedvolatilities. ..............................................................................369 15.12Dividends..... .....................................................................................371
Chapter16.Employeestockoptions....
16.1Contractualarrangements.. ...................................................................383
16.2Dooptionsaligntheinterestsofshareholdersandmanagers?... .................385
16.3Accountingissues. ..............................................................................386
16.4Valuation......
16.5Backdatingscandals.....
Chapter17.Optionsonstockindicesandcurrencies ............................................................396
17.1Optionsonstockindices... ...................................................................396
17.2Currencyoptions.. ..............................................................................398
17.3Optionsonstockspayingknowndividendyields.. ...................................401
17.4ValuationofEuropeanstockindexoptions... ..........................................403
17.5ValuationofEuropeancurrencyoptions... ..............................................406
17.6Americanoptions. ..............................................................................407
Chapter18.FuturesoptionsandBlack’smodel...... .............................................................412
18.1Natureoffuturesoptions ....................................................................412
18.2Reasonsforthepopularityoffuturesoptions..... ....................................415
18.3Europeanspotandfuturesoptions. ......................................................415
18.4Put–callparity.... ...............................................................................416
18.5Boundsforfuturesoptions..... .............................................................417
18.6Driftofafuturespricesinarisk-neutralworld..
18.7Black’smodelforvaluingfuturesoptions... ...........................................419
18.8UsingBlack’smodelinsteadofBlack–Scholes–Merton... .........................420
18.9Valuationoffuturesoptionsusingbinomialtrees ....................................421
18.10Americanfuturesoptionsvs.Americanspotoptions..
18.11Futures-styleoptions ...........................................................................424
Chapter19.TheGreekletters...
19.3Greeklettercalculation... ....................................................................431 19.4Deltahedging.....
19.7Relationshipbetweendelta,theta,andgamma....
19.10ApproximationforCalculationsoftheGreeksforanAmericanOption.
19.12Scenarioanalysis. ...............................................................................449
19.13Extensionofformulas.....
Chapter20.Volatilitysmiles.....
20.1Whythevolatilitysmileisthesameforcallsandputs
20.4Alternativewaysofcharacterizingthevolatilitysmile..
20.5Thevolatilitytermstructureandvolatilitysurfaces.....
20.6Minimumvariancedelta. ....................................................................471
20.7Theroleofthemodel.....
20.8Whenasinglelargejumpisanticipated......
Furtherquestions.. ..............................................................................476
Appendix:Determiningimpliedrisk-neutraldistributionsfrom volatilitysmiles. ...................................................................478
Chapter21.Basicnumericalprocedures...... .......................................................................481
21.1Binomialtrees...... ..............................................................................481
21.2Usingthebinomialtreeforoptionsonindices,currencies,andfutures contracts ............................................................................................489
21.3Binomialmodelforadividend-payingstock. ..........................................491
21.4Alternativeproceduresforconstructingtrees. ..........................................496
21.5Time-dependentparameters...... ............................................................499
21.6MonteCarlosimulation .......................................................................500
21.7Variancereductionprocedures.. ............................................................506
21.8Finitedifferencemethods.. ...................................................................509 Summary...... .....................................................................................519 Furtherreading.... ..............................................................................520 Practicequestions.. ..............................................................................521 Furtherquestions.. ..............................................................................523
Chapter22.Valueatriskandexpectedshortfall..... ............................................................525
22.1TheVaRandESmeasures ...................................................................526
22.2Historicalsimulation.... .......................................................................527
22.3Model-buildingapproach.. ...................................................................533
22.4Thelinearmodel.. ..............................................................................536
22.5Thequadraticmodel.... .......................................................................541
22.6MonteCarlosimulation .......................................................................544
22.7Comparisonofapproaches ...................................................................545
22.8Backtesting.. .....................................................................................545
22.9Principalcomponentsanalysis... ............................................................545 Summary...... .....................................................................................549 Furtherreading.... ..............................................................................550 Practicequestions.. ..............................................................................551 Furtherquestions.. ..............................................................................552
Chapter23.Estimatingvolatilitiesandcorrelations.. ............................................................554
23.1Estimatingvolatility..... .......................................................................554
23.2Theexponentiallyweightedmovingaveragemodel ...................................556
23.3TheGARCH(1,1)model. ...................................................................558
23.4Choosingbetweenthemodels... ............................................................559
23.5Maximumlikelihoodmethods... ............................................................560
23.6UsingGARCH(1,1)toforecastfuturevolatility.. ...................................565
23.7Correlations.. .....................................................................................568
23.8ApplicationofEWMAtofour-indexexample...... ...................................571 Summary...... .....................................................................................573 Furtherreading.... ..............................................................................573 Practicequestions.. ..............................................................................573 Furtherquestions.. ..............................................................................575
Chapter24.Creditrisk... ................................................................................................577
24.1Creditratings .....................................................................................577
24.2Historicaldefaultprobabilities.. ............................................................578
24.3Recoveryrates...... ..............................................................................579
24.4Estimatingdefaultprobabilitiesfrombondyieldspreads...... .....................580
24.5Comparisonofdefaultprobabilityestimates.. ..........................................583
24.6Usingequitypricestoestimatedefaultprobabilities..... ............................586
24.7Creditriskinderivativestransactions.. ..................................................588
24.8Defaultcorrelation.. ...........................................................................594
24.9CreditVaR. ......................................................................................597 Summary.... ......................................................................................600 Furtherreading... ...............................................................................601 Practicequestions ...............................................................................601 Furtherquestions ...............................................................................603
Chapter25.Creditderivatives... ......................................................................................605
25.1Creditdefaultswaps ...........................................................................606
25.2Valuationofcreditdefaultswaps.... ......................................................610
25.3Creditindices..... ...............................................................................613
25.4Theuseoffixedcoupons. ....................................................................614
25.5CDSforwardsandoptions..... .............................................................615
25.6Basketcreditdefaultswaps..... .............................................................616
25.7Totalreturnswaps.. ...........................................................................616
25.8Collateralizeddebtobligations. .............................................................617
25.9RoleofcorrelationinabasketCDSandCDO... ....................................619
25.10ValuationofasyntheticCDO. .............................................................620
25.11Alternativestothestandardmarketmodel..
Chapter26.Exoticoptions
26.1Packages.....
26.2PerpetualAmericancallandputoptions....
26.3NonstandardAmericanoptions...... ......................................................635
26.4Gapoptions
26.5Forwardstartoptions.....
26.6Cliquetoptions...
26.7Compoundoptions.. ...........................................................................637
26.8Chooseroptions.. ...............................................................................638
26.9Barrieroptions...
26.10Binaryoptions....
26.11Lookbackoptions...
26.12Shoutoptions.....
26.13Asianoptions.....
26.14Optionstoexchangeoneassetforanother..
26.15Optionsinvolvingseveralassets......
26.16Volatilityandvarianceswaps..
26.17Staticoptionsreplication.
Chapter27.Moreonmodelsandnumericalprocedures..
27.1AlternativestoBlack–Scholes–Merton.
27.2Stochasticvolatilitymodels.....
27.3TheIVFmodel... ...............................................................................668
27.4Convertiblebonds...
27.5Path-dependentderivatives......
27.6Barrieroptions..... ..............................................................................675
27.7Optionsontwocorrelatedassets...... .....................................................678
27.8MonteCarlosimulationandAmericanoptions.... ...................................680
Chapter28.Martingalesandmeasures ..............................................................................690
28.1Themarketpriceofrisk... ...................................................................691
28.2Severalstatevariables... .......................................................................694
28.3Martingales... .....................................................................................695
28.4Alternativechoicesforthenumeraire .....................................................696
28.5Extensiontoseveralfactors...... ............................................................699
28.6Black’smodelrevisited. .......................................................................700
28.7Optiontoexchangeoneassetforanother. ..............................................701
28.8Changeofnumeraire.... .......................................................................702
Chapter29.Interestratederivatives:Thestandardmarketmodels ..........................................708
29.1Bondoptions .....................................................................................708
29.2Interestratecapsandfloors...... ............................................................713
29.3Europeanswapoptions. .......................................................................719
29.4Hedginginterestratederivatives ............................................................723
Chapter30.Convexity,timing,andquantoadjustments... .....................................................728
30.1Convexityadjustments.. .......................................................................728
30.2Timingadjustments...... .......................................................................732
30.3Quantos ............................................................................................734
Chapter31.Equilibriummodelsoftheshortrate..... ............................................................741
31.1Background.. .....................................................................................741
31.2One-factormodels. ..............................................................................743
31.3Real-worldvs.risk-neutralprocesses. .....................................................748
31.4Estimatingparameters.. .......................................................................749
31.5Moresophisticatedmodels ...................................................................750
Chapter32.No-arbitragemodelsoftheshortrate... ............................................................754
32.1Extensionsofequilibriummodels..... .....................................................754
32.2Optionsonbonds. ..............................................................................758
32.3Volatilitystructures.. ...........................................................................759
32.4Interestratetrees. ...............................................................................760
32.5Ageneraltree-buildingprocedure... ......................................................762
32.6Calibration.. ......................................................................................771
32.7Hedgingusingaone-factormodel.. ......................................................773
Chapter33.HJM,LMM,andmultiplezerocurves...... ......................................................777
33.1TheHeath,Jarrow,andMortonmodel...... ...........................................777
33.2TheLIBORmarketmodel..... .............................................................780
33.3Handlingmultiplezerocurves. .............................................................790
33.4Agencymortgage-backedsecurities. ......................................................791
Chapter34.SwapsRevisited..... ......................................................................................796
34.1Variationsonthevanilladeal. .............................................................796
34.2Compoundingswaps ...........................................................................798
34.3Currencyswaps... ...............................................................................799
34.4Morecomplexswaps......
34.5Equityswaps...... ...............................................................................803
34.6Swapswithembeddedoptions. .............................................................804 34.7Otherswaps ......................................................................................807
Chapter35.Energyandcommodityderivatives...... .............................................................811
35.1Agriculturalcommodities ....................................................................812
35.2Metals. .............................................................................................812
35.3Energyproducts.. ...............................................................................813
35.4Modelingcommodityprices.... .............................................................815
35.5Weatherderivatives.. ...........................................................................821
35.6Insurancederivatives ...........................................................................822
35.7Pricingweatherandinsurancederivatives.... ...........................................824
35.8Howanenergyproducercanhedgerisks....
Chapter36.Realoptions... .............................................................................................829
36.1Capitalinvestmentappraisal... .............................................................829
36.2Extensionoftherisk-neutralvaluationframework..... .............................830
36.3Estimatingthemarketpriceofrisk. ......................................................832
36.4Applicationtothevaluationofabusiness.. ...........................................833
36.5Evaluatingoptionsinaninvestmentopportunity
37.1Lessonsforallusersofderivatives....
37.2Lessonsforfinancialinstitutions......
37.3Lessonsfornonfinancialcorporations......
TECHNICALNOTES
AvailableontheAuthor’sWebsite http://www-2.rotman.utoronto.ca/ hull/TechnicalNotes/index.html
1.ConvexityAdjustmentstoEurodollarFutures
2.PropertiesoftheLognormalDistribution
3.WarrantValuationWhenValueofEquityplusWarrantsIsLognormal
4.ExactProcedureforValuingAmericanCallsonStocksPayingaSingleDividend
5.CalculationoftheCumulativeProbabilityinaBivariateNormalDistribution
6.DifferentialEquationforPriceofaDerivativeonaStockPayingaKnownDividend Yield
7.DifferentialEquationforPriceofaDerivativeonaFuturesPrice
8.AnalyticApproximationforValuingAmericanOptions
9.GeneralizedTree-BuildingProcedure
10.TheCornish–FisherExpansiontoEstimateVaR
11.ManipulationofCreditTransitionMatrices
12.CalculationofCumulativeNoncentralChi-SquareDistribution
13.EfficientProcedureforValuingAmerican-StyleLookbackOptions
14.TheHull–WhiteTwo-FactorModel
15.ValuingOptionsonCoupon-BearingBondsinaOne-FactorInterestRateModel
16.ConstructionofanInterestRateTreewithNonconstantTimeStepsandNonconstant Parameters
17.TheProcessfortheShortRateinanHJMTermStructureModel
18.ValuationofaCompoundingSwap
19.ValuationofanEquitySwap
20.ChangingtheMarketPriceofRiskforVariablesThatAreNotthePricesofTraded Securities
21.HermitePolynomialsandTheirUseforIntegration
22.ValuationofaVarianceSwap
23.TheBlack,Derman,ToyModel
24.ProofthatForwardandFuturesPricesareEqualWhenInterestRatesAreConstant
25.ACash-FlowMappingProcedure
26.ABinomialMeasureofCreditCorrelation
27.CalculationofMomentsforValuingAsianOptions
28.CalculationofMomentsforValuingBasketOptions
29.ProofofExtensionstoIto’sLemma
30.TheReturnofaSecurityDependentonMultipleSourcesofUncertainty
31.PropertiesofHo–LeeandHull–WhiteInterestRateModels
Preface
Itissometimeshardtobelievethatthefirsteditionofthisbookwasonly330pagesand 13chapterslong!Thebookhasgrownandbeenadaptedtokeepupwiththefastpace ofchangeinderivativesmarkets.
Likeearliereditions,thebookservesseveralmarkets.Itisappropriateforgraduate coursesinbusiness,economics,financialmathematics,andfinancialengineering.Itcan beusedonadvancedundergraduatecourseswhenstudentshavegoodquantitative skills.Also,manypractitionerswhoareinvolvedinderivativesmarketsfindthebook useful.Wearedelightedthathalfthepurchasersofthebookareanalysts,traders,and otherderivativesprofessionals.
Oneofthekeydecisionsthatmustbemadebyanauthorwhoiswritingintheareaof derivativesconcernstheuseofmathematics.Ifthelevelofmathematicalsophistication istoohigh,thematerialislikelytobeinaccessibletomanystudentsandpractitioners.If itistoolow,someimportantissueswillinevitablybetreatedinarathersuperficialway. Wehavetriedtobeparticularlycarefulaboutthewaybothmathematicsandnotation havebeenusedinthebook.Nonessentialmathematicalmaterialhasbeeneither eliminatedorincludedinend-of-chapterappendicesorintechnicalnotesonthewebsite www.rotman.utoronto.ca/ hull .Conceptsthatarelikelytobenewtomanyreaders havebeenexplainedcarefully,andmanynumericalexampleshavebeenincluded.
Thebookassumesthatthereaderhastakenanintroductorycourseinfinanceandan introductorycourseinprobabilityandstatistics.Nopriorknowledgeofoptions, futurescontracts,swaps,andsoon,isassumed.Itisnotthereforenecessaryfor studentstotakeanelectivecourseininvestmentspriortotakingacoursebasedon thisbook.
Options,FuturesandOtherDerivatives canbeusedforafirstcourseinderivativesor foramoreadvancedcourse.Therearemanydifferentwaysitcanbeusedinthe classroom.Instructorsteachingafirstcourseinderivativesarelikelytowanttospend mostclassroomtimeonthefirsthalfofthebook.Instructorsteachingamoreadvanced coursewillfindthatmanydifferentcombinationsofchaptersinthesecondhalfofthe bookcanbeused.ThematerialinChapter37workswellattheendofeitheran introductoryoranadvancedcourse.
What ’sNewintheTenthEdition?
Materialhasbeenupdatedandimproved.OISdiscountingisnowusedthroughoutthe book.Thismakesthepresentationofthematerialmorestraightforwardandmore theoreticallyappealing.Thevaluationofinstrumentssuchasswapsandforwardrate agreementsrequires(a)forwardratesfortherateusedtocalculatepayments(usually LIBOR)and(b)therisk-freezerocurveusedfordiscounting(usuallytheOISzero
curve).Themethodspresentedcanbeextendedtosituationswherepaymentsare dependentonanyriskyrate.
Thechangesinthetentheditionincludethefollowing:
1. Arewriteofthechapteronswaps(Chapter7)toimprovepresentationand reflectchangingmarketpractices.
2. Anewchapter(Chapter9)onvaluationadjustments(CVA,DVA,FVA,MVA, andKVA).FinancialeconomistshavereservationsaboutFVA,MVA,andKVA (andtheseareexplained),butXVAshavebecomesuchanimportantpartof derivativesvaluationthatitisimportanttocoverthem.
3. Materialatvariouspointsinthebookonhownegativeinterestratescanbe handledinpricingmodels.Intheno-arbitrageworldthatweassumewhenvaluing derivatives,negativeratesmakenosense.Buttheyareafeatureoffinancial marketsinanumberofEuropeancountriesandJapanandcannotbeignored.
4. Anewchapteronequilibriummodelsofthetermstructure(Chapter31).These modelsareimportantpedagogicallyandarewidelyusedinlong-termscenario analyses.Idecidedthattheydeservedtheirownchapter.
5. MoredetailsonthecalculationofGreeklettersandsmiledynamics.
6. Morediscussionoftheexpectedshortfallmeasureandstressedriskmeasures, reflectingtheirincreasinguseinregulationandriskmanagement.
7. CoverageoftheSABRmodel.
8. UpdatedmaterialonCCPsandtheregulationofOTCderivatives.
9. Improvedmaterialonmartingalesandmeasures,tailingthehedge,bootstrap methods,andconvertiblebonds.
10. Updatingofexamplestoreflectcurrentmarketconditions.
11. Newend-ofchapterproblemsandrevisionstomanyoldend-of-chapterproblems.
12. NewversionofthesoftwareDerivaGem.
Software
DerivaGem4.00isincludedwiththisbook.Asbefore,thisconsistsoftwoExcel applications:the OptionsCalculator andthe ApplicationsBuilder.TheOptionsCalculator consistsofeasy-to-usesoftwareforvaluingawiderangeofoptions.TheApplications BuilderconsistsofanumberofExcelfunctionsfromwhichuserscanbuildtheirown applications.Itincludesanumberofsampleapplicationsandenablesstudentstoexplore thepropertiesofoptionsandnumericalproceduresmoreeasily.Italsoallowsmore interestingassignmentstobedesigned.
DerivaGem4.00allowsanumberofnewmodels(Heston,SABR,Bacheliernormal, anddisplacedlognormal)tobeusedforvaluation.Thesoftwareisdescribedmorefully attheendofthebook.Youcandownloadthesoftwarefrom: www.pearsoned.co.in/hull10e.
Acknowledgments
Manypeoplehaveplayedapartinthedevelopmentofsuccessiveeditionsofthisbook. Indeed,thelistofpeoplewhohaveprovidedmewithfeedbackonthebookisnowso longthatitisnotpossibletomentioneveryone.Wehavebenefitedfromtheadviceof manyacademicswhohavetaughtfromthebookandfromthecommentsofmany derivativespractitioners.Inparticular,wewouldliketothankthestudentsonour coursesattheUniversityofTorontoandattheIndianInstituteofManagement Bangalorewhohavemademanysuggestionsonhowthematerialcanbeimproved. EddieMizzifromTheGeometricPressdidanexcellentjobeditingthefinalmanuscript andhandlingpagecomposition.EmilioBaronefromLuissGuidoCarliUniversityin Rome,PrakashApteandMRRaofromIndianInstituteofManagementBangalore andIndianSchoolofBusinessHyderabadprovidedmanydetailedcomments.
AlanWhite,acolleagueattheUniversityofToronto,deservesaspecialacknowledgment.AlanandJohnhasbeencarryingoutjointresearchandconsultingintheareasof derivativesandriskmanagementforover30years,spendingalotoftimediscussingkey issues;alotusandme(Sankarshan)includedhavestartedlearningaboutDerivatives fromworksofAlanandJohn.Manyofthenewideasinthisbook,andmanyofthenew waysusedtoexplainoldideas,areasmuchAlan’sasours.Alanhasdonemostofthe developmentworkontheDerivaGemsoftware.
SpecialthanksareduetomanypeopleatPearson,particularlyDonnaBattista, NeerajBhalla,NicoleSuddeth,AlisonKalil,PradeepKumarBhattacharjee,and Purushothaman,C.fortheirenthusiasm,adviceandencouragement.
Wewelcomecommentsonthebookfromreaders.Oure-mailaddressare:
hull@rotman.utoronto.ca Sankarshan.basu@iimb.ac.in
JohnHull SankarshanBasu
Introduction
Inthelast40years,derivativeshavebecomeincreasinglyimportantinfinance.Futures andoptionsareactivelytradedonmanyexchangesthroughouttheworld.Many differenttypesofforwardcontracts,swaps,options,andotherderivativesareentered intobyfinancialinstitutions,fundmanagers,andcorporatetreasurersintheover-thecountermarket.Derivativesareaddedtobondissues,usedinexecutivecompensation plans,embeddedincapitalinvestmentopportunities,usedtotransferrisksinmortgages fromtheoriginallenderstoinvestors,andsoon.Wehavenowreachedthestagewhere thosewhoworkinfinance,andmanywhoworkoutsidefinance,needtounderstand howderivativeswork,howtheyareused,andhowtheyarepriced.
Whetheryoulovederivativesorhatethem,youcannotignorethem!Thederivatives marketishuge—muchbiggerthanthestockmarketwhenmeasuredintermsof underlyingassets.Thevalueoftheassetsunderlyingoutstandingderivativestransactionsisseveraltimestheworldgrossdomesticproduct.Asweshallseeinthischapter, derivativescanbeusedforhedgingorspeculationorarbitrage.Theycanbeusedto transferawiderangeofrisksintheeconomyfromoneentitytoanother.
A derivative canbedefinedasafinancialinstrumentwhosevaluedependson(or derivesfrom)thevaluesofother,morebasic,underlyingvariables.Veryoftenthe variablesunderlyingderivativesarethepricesoftradedassets.Astockoption,for example,isaderivativewhosevalueisdependentonthepriceofastock.However, derivativescanbedependentonalmostanyvariable,fromthepriceofhogstothe amountofsnowfallingatacertainskiresort.
Sincethefirsteditionofthisbookwaspublishedin1988therehavebeenmany developmentsinderivativesmarkets.Thereisnowactivetradingincreditderivatives, electricityderivatives,weatherderivatives,andinsurancederivatives.Manynewtypes ofinterestrate,foreignexchange,andequityderivativeproductshavebeencreated. Therehavebeenmanynewideasinriskmanagementandriskmeasurement.Capital investmentappraisalnowofteninvolvestheevaluationofwhatareknownas real options.Manynewregulationshavebeenintroducedcoveringover-the-counterderivativesmarkets.Thebookhaskeptupwithallthesedevelopments.
Derivativesmarketshavecomeunderagreatdealofcriticismbecauseoftheirrolein thecreditcrisisthatstartedin2007.Derivativeproductswerecreatedfromportfoliosof riskymortgagesintheUnitedStatesusingaprocedureknownassecuritization.Manyof theproductsthatwerecreatedbecameworthlesswhenhousepricesdeclined.Financial
institutions,andinvestorsthroughouttheworld,lostahugeamountofmoneyandthe worldwasplungedintotheworstrecessionithadexperiencedin75years.Chapter8 explainshowsecuritizationworksandwhysuchbiglossesoccurred.
Thewaymarketparticipantstradeandvaluederivativeshasevolvedthroughtime. Regulatoryrequirementsintroducedsincethecrisishavehadahugeeffectontheoverthe-countermarket.Collateralandcreditissuesarenowgivenmuchmoreattention thaninthepast.
Marketparticipantshavechangedtheproxytheyusefortherisk-freerate.Theyalso nowcalculateanumberofvaluationadjustmentstoreflectfundingcostsandcapital requirements,aswellascreditrisk.Thiseditionhasbeenchangedtokeepuptodate withthesedevelopments.Chapter9isnowdevotedtoadiscussionofhowvaluation adjustmentsworkandtheextenttowhichtheyaretheoreticallyvalid.
Inthisopeningchapter,wetakeafirstlookatderivativesmarketsandhowtheyare changing.Wedescribeforward,futures,andoptionsmarketsandprovideanoverview ofhowtheyareusedbyhedgers,speculators,andarbitrageurs.Laterchapterswillgive moredetailsandelaborateonmanyofthepointsmadehere.
1.1EXCHANGE-TRADEDMARKETS
Aderivativesexchangeisamarketwhereindividualstradestandardizedcontractsthat havebeendefinedbytheexchange.Derivativesexchangeshaveexistedforalongtime. TheChicagoBoardofTrade(CBOT)wasestablishedin1848tobringfarmersand merchantstogether.Initiallyitsmaintaskwastostandardizethequantitiesand qualitiesofthegrainsthatweretraded.Withinafewyears,thefirstfutures-type contractwasdeveloped.Itwasknownasa to-arrivecontract.Speculatorssoonbecame interestedinthecontractandfoundtradingthecontracttobeanattractivealternative totradingthegrainitself.Arivalfuturesexchange,theChicagoMercantileExchange (CME),wasestablishedin1919.Nowfuturesexchangesexistallovertheworld.(See tableattheendofthebook.)TheCMEandCBOThavemergedtoformthe CMEGroup(www.cmegroup.com),whichalsoincludestheNewYorkMercantile Exchange(NYMEX),andtheKansasCityBoardofTrade(KCBT).
TheChicagoBoardOptionsExchange(CBOE, www.cboe.com )startedtradingcall optioncontractson16stocksin1973.Optionshadtradedpriorto1973,buttheCBOE succeededincreatinganorderlymarketwithwell-definedcontracts.Putoption contractsstartedtradingontheexchangein1977.TheCBOEnowtradesoptionson thousandsofstocksandmanydifferentstockindices.Likefutures,optionshaveproved tobeverypopularcontracts.Manyotherexchangesthroughouttheworldnowtrade options.(Seetableattheendofthebook.)Theunderlyingassetsincludeforeign currenciesandfuturescontractsaswellasstocksandstockindices.
Oncetwotradershaveagreedonatrade,itishandledbytheexchangeclearing house.Thisstandsbetweenthetwotradersandmanagestherisks.Suppose,for example,thattraderAagreestobuy100ouncesofgoldfromtraderBatafuture timefor$1,250perounce.TheresultofthistradewillbethatAhasacontracttobuy 100ouncesofgoldfromtheclearinghouseat$1,250perounceandBhasacontractto sell100ouncesofgoldtotheclearinghousefor$1,250perounce.Theadvantageof thisarrangementisthattradersdonothavetoworryaboutthecreditworthinessofthe peopletheyaretradingwith.Theclearinghousetakescareofcreditriskbyrequiring
eachofthetwotraderstodepositfunds(knownasmargin)withtheclearinghouseto ensurethattheywillliveuptotheirobligations.Marginrequirementsandtheoperation ofclearinghousesarediscussedinmoredetailinChapter2.
ElectronicMarkets
Traditionallyderivativesexchangeshaveusedwhatisknownasthe openoutcrysystem Thisinvolvestradersphysicallymeetingontheflooroftheexchange,shouting,and usingacomplicatedsetofhandsignalstoindicatethetradestheywouldliketocarry out.Exchangeshavelargelyreplacedtheopenoutcrysystemby electronictrading.This involvestradersenteringtheirdesiredtradesatakeyboardandacomputerbeingused tomatchbuyersandsellers.Theopenoutcrysystemhasitsadvocates,but,astime passes,itisbecominglessandlessused.
Electronictradinghasledtoagrowthinhigh-frequencyandalgorithmictrading. Thisinvolvestheuseofcomputerprogramstoinitiatetrades,oftenwithouthuman intervention,andhasbecomeanimportantfeatureofderivativesmarkets.
1.2OVER-THE-COUNTERMARKETS
Notallderivativestradingisonexchanges.Manytradestakeplaceinthe over-thecounter (OTC)market.Banks,otherlargefinancialinstitutions,fundmanagers,and corporationsarethemainparticipantsinOTCderivativesmarkets.OnceanOTC tradehasbeenagreed,thetwopartiescaneitherpresentittoacentralcounterparty (CCP)orclearthetradebilaterally.ACCPislikeanexchangeclearinghouse.It standsbetweenthetwopartiestothederivativestransactionsothatonepartydoesnot havetobeartheriskthattheotherpartywilldefault.Whentradesarecleared bilaterally,thetwopartieshaveusuallysignedanagreementcoveringalltheirtransactionswitheachother.Theissuescoveredintheagreementincludethecircumstances underwhichoutstandingtransactionscanbeterminated,howsettlementamountsare calculatedintheeventofatermination,andhowthecollateral(ifany)thatmustbe postedbyeachsideiscalculated.CCPsandbilateralclearingarediscussedinmore detailinChapter2.
Largebanksoftenactasmarketmakersforthemorecommonlytradedinstruments. Thismeansthattheyarealwayspreparedtoquoteabidprice(atwhichtheyare preparedtotakeonesideofaderivativestransaction)andanofferprice(atwhichthey arepreparedtotaketheotherside).
Priortothecreditcrisis,whichstartedin2007andisdiscussedinsomedetailin Chapter8,OTCderivativesmarketswerelargelyunregulated.Followingthecredit crisisandthefailureofLehmanBrothers(seeBusinessSnapshot1.1),wehaveseenthe developmentofmanynewregulationsaffectingtheoperationofOTCmarkets.The mainobjectivesoftheregulationsaretoimprovethetransparencyofOTCmarketsand reducesystemicrisk(seeBusinessSnapshot1.2).Theover-the-countermarketinsome respectsisbeingforcedtobecomemoreliketheexchange-tradedmarket.Three importantchangesare:
1. StandardizedOTCderivativesbetweentwofinancialinstitutionsintheUnited Statesmust,wheneverpossible,betradedonwhatarereferredtoa swapexecution
BusinessSnapshot1.1 TheLehmanBankruptcy
OnSeptember15,2008,LehmanBrothersfiledforbankruptcy.Thiswasthelargest bankruptcyinU.S.historyanditsramificationswerefeltthroughoutderivatives markets.Almostuntiltheend,itseemedasthoughtherewasagoodchancethat Lehmanwouldsurvive.Anumberofcompanies(e.g.,theKoreanDevelopment Bank,BarclaysBankintheUnitedKingdom,andBankofAmerica)expressed interestinbuyingit,butnoneofthesewasabletocloseadeal.Manypeoplethought thatLehmanwas‘‘toobigtofail’’andthattheU.S.governmentwouldhavetobailit outifnopurchasercouldbefound.Thisprovednottobethecase.
Howdidthishappen?Itwasacombinationofhighleverage,riskyinvestments,and liquidityproblems.Commercialbanksthattakedepositsaresubjecttoregulationson theamountofcapitaltheymustkeep.Lehmanwasaninvestmentbankandnot subjecttotheseregulations.By2007,itsleverageratiohadincreasedto31:1,which meansthata3–4%declineinthevalueofitsassetswouldwipeoutitscapital.Dick Fuld,Lehman’sChairmanandChiefExecutiveOfficer,encouragedanaggressive deal-making,risk-takingculture.Heisreportedtohavetoldhisexecutives:‘‘Every dayisabattle.Youhavetokilltheenemy.’’TheChiefRiskOfficeratLehmanwas competent,butdidnothavemuchinfluenceandwasevenremovedfromtheexecutive committeein2007.TheriskstakenbyLehmanincludedlargepositionsinthe instrumentscreatedfromsubprimemortgages,whichwillbedescribedinChapter8. Lehmanfundedmuchofitsoperationswithshort-termdebt.Whentherewasaloss ofconfidenceinthecompany,lendersrefusedtorenewthisfunding,forcingitinto bankruptcy.
Lehmanwasveryactiveintheover-the-counterderivativesmarkets.Ithadovera milliontransactionsoutstandingwithabout8,000differentcounterparties. Lehman’scounterpartieswereoftenrequiredtopostcollateralandthiscollateral hadinmanycasesbeenusedbyLehmanforvariouspurposes.Litigationaimedat determiningwhooweswhattowhomcontinuedformanyyearsafterthebankruptcyfiling.
facilities (SEFs).Theseareplatformssimilartoexchangeswheremarket participantscanpostbidandofferquotesandwheremarketparticipantscan tradebyacceptingthequotesofothermarketparticipants.
2. ThereisarequirementinmostpartsoftheworldthataCCPbeusedformost standardizedderivativestransactionsbetweenfinancialinstitutions.
3. Alltradesmustbereportedtoacentralrepository.
MarketSize
Boththeover-the-counterandtheexchange-tradedmarketforderivativesarehuge.The numberofderivativestransactionsperyearinOTCmarketsissmallerthaninexchangetradedmarkets,buttheaveragesizeofthetransactionsismuchgreater.Althoughthe statisticsthatarecollectedforthetwomarketsarenotexactlycomparable,itisclearthat thevolumeofbusinessintheover-the-countermarketismuchlargerthaninthe exchange-tradedmarket.TheBankforInternationalSettlements(www.bis.org)started collectingstatisticsonthemarketsin1998.Figure1.1compares(a)theestimatedtotal
BusinessSnapshot1.2 SystemicRisk
Systemicriskistheriskthatadefaultbyonefinancialinstitutionwillcreatea‘‘ripple effect’’thatleadstodefaultsbyotherfinancialinstitutionsandthreatensthestability ofthefinancialsystem.Therearehugenumbersofover-the-countertransactions betweenbanks.IfBankAfails,BankBmaytakeahugelossonthetransactionsit haswithBankA.ThisinturncouldleadtoBankBfailing.BankCthathasmany outstandingtransactionswithbothBankAandBankBmightthentakealargeloss andexperienceseverefinancialdifficulties;andsoon.
ThefinancialsystemhassurviveddefaultssuchasDrexelin1990andLehman Brothersin2008,butregulatorscontinuetobeconcerned.Duringthemarketturmoil of2007and2008,manylargefinancialinstitutionswerebailedout,ratherthanbeing allowedtofail,becausegovernmentswereconcernedaboutsystemicrisk.
principalamountsunderlyingtransactionsthatwereoutstandingintheover-thecounter marketsbetweenJune1998andDecember2015and(b)theestimatedtotalvalueofthe assetsunderlyingexchange-tradedcontractsduringthesameperiod.Usingthese measures,thesizeoftheover-the-countermarketinDecember2015was$492.9trillion andthesizeoftheexchange-tradedmarketwas$63.3trillion.1
Figure1.1showsthatthe OTCmarketgrewrapidlyupto2007,buthasseenverylittlenetgrowthsincethen.One reasonforthelackofgrowthisthepopularityof compression.Thisisaprocedurewhere twoormorecounterpartiesrestructuretransactionswitheachotherwiththeresultthat theunderlyingprincipalisreduced.
IninterpretingFigure1.1,weshouldbearinmindthattheprincipalunderlyingan over-the-countertransactionisnotthesameasitsvalue.Anexampleofanover-thecountertransactionisanagreementtobuy100millionU.S.dollarswithBritishpounds
Figure1.1 Sizeofover-the-counterandexchange-tradedderivativesmarkets.
1 WhenaCCPstandsbetweentwosidesinanOTCtransaction,twotransactionsareconsideredtohave beencreatedforthepurposesoftheBISstatistics.