Credit Derivatives Pricing Models: Models, Pricing and Implementation

400 Pages·2003·1.791 MB·English

by  Philipp J. Schönbucher

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Credit Derivatives Pricing Models
Models, Pricing and Implementation
Philipp J. Scho¨nbucher
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Credit Derivatives Pricing Models
i
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ii
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Credit Derivatives Pricing Models
Models, Pricing and Implementation
Philipp J. Scho¨nbucher
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Contents
Preface xi
Acknowledgements xv
Abbreviations xvii
Notation xix
1 Introduction 1
1.1 Theworldofcreditrisk 1
1.2 Thecomponentsofcreditrisk 2
1.3 Marketstructure 4
2 CreditDerivatives:OverviewandHedge-BasedPricing 7
2.1 Theemergenceofanewclassofderivatives 7
2.2 Terminology 7
2.3 Underlyingassets 10
2.3.1 Loans 10
2.3.2 Bonds 11
2.3.3 Convertiblebonds 12
2.3.4 Counterpartyrisk 12
2.4 Assetswaps 12
2.5 Totalreturnswaps 13
2.6 Creditdefaultswaps 15
2.7 Hedge-basedpricing 19
2.7.1 Hedgeinstruments 20
2.7.2 Shortpositionsindefaultablebonds 20
2.7.3 Assetswappackages 22
2.7.4 Totalreturnswaps 25
2.7.5 Creditdefaultswaps 27
2.8 Exoticcreditderivatives 37
2.8.1 Defaultdigitalswaps 37
2.8.2 Exoticdefaultpaymentsincreditdefaultswaps 38
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vi Contents
2.8.3 Rating-triggeredcreditdefaultswaps 39
2.8.4 Optionsondefaultablebonds 40
2.8.5 Creditspreadoptions 41
2.9 DefaultcorrelationproductsandCDOs 43
2.9.1 First-to-defaultswapsandbasketdefaultswaps 43
2.9.2 Firstlosslayers 44
2.9.3 Collateraliseddebtobligations 46
2.10 Credit-linkednotes 49
2.11 Guidetotheliterature 50
3 CreditSpreadsandBondPrice-BasedPricing 51
3.1 Creditspreadsandimplieddefaultprobabilities 52
3.1.1 Risk-neutralprobabilities 52
3.1.2 Setup 52
3.1.3 Thefundamentalrelationship 54
3.1.4 Theimpliedsurvivalprobability 54
3.1.5 Conditionalsurvivalprobabilitiesandimplied
hazardrates 56
3.1.6 Relationtoforwardspreads 58
3.2 Recoverymodelling 60
3.3 Buildingblocksforcreditderivativespricing 61
3.4 Pricingwiththebuildingblocks 64
3.4.1 Defaultablefixed-couponbond 64
3.4.2 Defaultablefloater 65
3.4.3 Variantsofcouponbonds 66
3.4.4 Creditdefaultswaps 66
3.4.5 ForwardstartCDSs 68
3.4.6 Defaultdigitalswaps 68
3.4.7 Assetswappackages 69
3.5 Constructingandcalibratingcreditspreadcurves 69
3.5.1 Parametricformsforthespreadcurves 70
3.5.2 Semi-parametricandnon-parametriccalibration 72
3.5.3 Approximativeandaggregatefits 74
3.5.4 Calibrationexample 75
3.6 Spreadcurves:issuesinimplementation 77
3.6.1 Whichdefault-freeinterestratesshouldoneuse? 77
3.6.2 Recoveryuncertainty 79
3.6.3 Buckethedging 81
3.7 Spreadcurves:discussion 82
3.8 Guidetotheliterature 83
4 MathematicalBackground 85
4.1 Stoppingtimes 86
4.2 Thehazardrate 87
4.3 Pointprocesses 88
4.4 Theintensity 88
4.5 Markedpointprocessesandthejumpmeasure 91
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Contents vii
4.6 Thecompensatormeasure 93
4.6.1 Randommeasuresindiscretetime 95
4.7 Examplesforcompensatormeasures 97
4.8 Itoˆ’slemmaforjumpprocesses 100
4.9 ApplicationsofItoˆ’slemma 101
4.9.1 Predictablecompensatorsforjumpprocesses 102
4.9.2 Itoˆ productruleandItoˆ quotientrule 103
4.9.3 Thestochasticexponential 104
4.10 Martingalemeasure,fundamentalpricingruleandincompleteness 105
4.11 Changeofnumeraireandpricingmeasure 107
4.11.1 TheRadon–Nikodymtheorem 107
4.11.2 TheGirsanovtheorem 108
4.12 Thechangeofmeasure/changeofnumerairetechnique 109
5 AdvancedCreditSpreadModels 111
5.1 Poissonprocesses 111
5.1.1 Amodelfordefaultarrivalrisk 111
5.1.2 IntuitiveconstructionofaPoissonprocess 112
5.1.3 PropertiesofPoissonprocesses 113
5.1.4 SpreadswithPoissonprocesses 115
5.2 InhomogeneousPoissonprocesses 115
5.2.1 Pricingthebuildingblocks 117
5.3 Stochasticcreditspreads 118
5.3.1 Coxprocesses 119
5.3.2 Pricingthebuildingblocks 125
5.3.3 Generalpointprocesses 126
5.3.4 CompoundPoissonprocesses 128
6 RecoveryModelling 131
6.1 Presentationofthedifferentrecoverymodels 132
6.1.1 Zerorecovery 132
6.1.2 Recoveryoftreasury 133
6.1.3 Multipledefaultsandrecoveryofmarketvalue 135
6.1.4 Recoveryofpar 141
6.1.5 Stochasticrecoveryandrecoveryrisk 143
6.1.6 Commonparametricdistributionfunctionsforrecoveries 147
6.1.7 ValuationofthedeliveryoptioninaCDS 148
6.2 Comparingtherecoverymodels 150
6.2.1 Theoreticalcomparisonoftherecoverymodels 150
6.2.2 Empiricalanalysisofrecoveryrates 159
7 ImplementationofIntensity-BasedModels 165
7.1 Tractablemodelsofthespotintensity 166
7.1.1 Thetwo-factorGaussianmodel 167
7.1.2 ThemultifactorGaussianmodel 171
7.1.3 Impliedsurvivalprobabilities 172
7.1.4 Payoffsatdefault 174