Pricing Interest-Rate Derivatives
A Fourier-Transform Based Approach
1Introduction ...............................................1
1.1MotivationandObjectives................................1
1.2StructureoftheThesis...................................4
2AGeneralMulti-FactorModeloftheTermStructure
ofInterestRatesandthePrinciplesofCharacteristic
Functions ..................................................7
2.1AnExtendedJump-Di?usionTerm-StructureModel.........7
2.2TechnicalPreliminaries...................................11
2.3TheRisk-NeutralPricingApproach........................13
2.3.1ArbitrageandtheEquivalentMartingaleMeasure.....15
2.3.2DerivationoftheRisk-NeutralCoe?cients............16
2.4TheCharacteristicFunction..............................21
3TheoreticalPricesofEuropeanInterest-RateDerivatives ..31
3.1Overview...............................................31
3.2DerivativeswithUnconditionalPayo?Functions.............32
3.3DerivativeswithConditionalPayo?Functions...............38
4ThreeFourierTransform-BasedPricingApproaches .......45
4.1Overview...............................................45
4.2HestonApproach........................................49
4.3Carr-MadanApproach...................................55
4.4LewisApproach.........................................60
5PayoTransformationsandthePricingofEuropean
Interest-RateDerivatives ..................................69
5.1Overview...............................................69
5.2UnconditionalPayo?Functions...........................70
5.2.1GeneralResults...................................70
5.2.2PricingUnconditionalInterest-RateContracts........79
5.3ConditionalPayo?Functions..............................81
5.3.1GeneralResults...................................82
5.3.2PricingofZero-BondOptionsandInterest-RateCaps
andFloors........................................87
5.3.3PricingofCoupon-BondOptionsandYield-Based
Swaptions........................................90
6NumericalComputationofModelPrices ..................95
6.1Overview...............................................95
6.2ContractswithUnconditionalExerciseRights...............96
6.3ContractswithConditionalExerciseRights.................97
6.3.1CalculatingOptionPriceswiththeIFFT.............97
6.3.2Re?nementoftheIFFTPricingAlgorithm...........101
6.3.3DeterminationoftheOptimalParametersforthe
NumericalScheme.................................103
7JumpSpecicationsforAneTerm-StructureModels.....111
7.1Overview...............................................111
7.2ExponentiallyDistributedJumps..........................115
7.3NormallyDistributedJumps..............................117
7.4GammaDistributedJumps...............................120
8Jump-EnhancedOne-FactorInterest-RateModels .........125
8.1Overview...............................................125
8.2TheOrnstein-UhlenbeckModel...........................126
ContentsXIII
8.2.1DerivationoftheCharacteristicFunction.............126
8.2.2NumericalResults.................................128
8.3TheSquare-RootModel..................................136
8.3.1DerivationoftheCharacteristicFunction.............136
8.3.2NumericalResults.................................138
9Jump-EnhancedTwo-FactorInterest-RateModels .........145
9.1Overview...............................................145
9.2TheAdditiveOU-SRModel..............................146
9.2.1DerivationoftheCharacteristicFunction.............146
9.2.2NumericalResults.................................148
9.3TheFong-VasicekModel.................................159
9.3.1DerivationoftheCharacteristicFunction.............159
9.3.2NumericalResults.................................163
10Non-AneTerm-StructureModelsandShort-Rate
ModelswithStochasticJumpIntensity ....................171
10.1Overview...............................................171
10.2QuadraticGaussianModels...............................171
10.3StochasticJumpIntensity................................174
11Conclusion ................................................175
ADerivationoftheComplex-ValuedCoecientsforthe
CharacteristicFunctionintheSquare-RootModel.........179
BDerivationoftheComplex-ValuedCoecientsforthe
CharacteristicFunctionintheFong-VasicekModel ........183
References.....................................................187