Pricing Interest-Rate Derivatives
A Fourier-TransformBased Approach


1 Introduction ............................................... 1
1.1 MotivationandObjectives................................ 1
1.2 Structureof theThesis................................... 4
2 A General Multi-Factor Model of the Term Structure
of Interest Rates and the Principles of Characteristic
Functions .................................................. 7
2.1 An Extended Jump-Di?usion Term-Structure Model . . . . . . . . . 7
2.2 TechnicalPreliminaries................................... 11
2.3 TheRisk-NeutralPricingApproach........................ 13
2.3.1 Arbitrage and the Equivalent Martingale Measure . . . . . 15
2.3.2 Derivationof theRisk-NeutralCoe?cients............ 16
2.4 TheCharacteristicFunction .............................. 21
3 Theoretical Prices of European Interest-Rate Derivatives .. 31
3.1 Overview............................................... 31
3.2 DerivativeswithUnconditionalPayo?Functions............. 32
3.3 DerivativeswithConditionalPayo?Functions............... 38
4 Three Fourier Transform-Based Pricing Approaches ....... 45
4.1 Overview............................................... 45
4.2 HestonApproach........................................ 49
4.3 Carr-MadanApproach ................................... 55
4.4 LewisApproach......................................... 60
5 Payo? Transformations and the Pricing of European
Interest-Rate Derivatives .................................. 69
5.1 Overview............................................... 69
5.2 UnconditionalPayo?Functions ........................... 70
5.2.1 GeneralResults ................................... 70
5.2.2 Pricing Unconditional Interest-Rate Contracts . . . . . . . . 79
5.3 ConditionalPayo?Functions.............................. 81
5.3.1 GeneralResults ................................... 82
5.3.2 Pricing of Zero-Bond Options and Interest-Rate Caps
andFloors........................................ 87
5.3.3 Pricing of Coupon-Bond Options and Yield-Based
Swaptions ........................................ 90
6 Numerical Computation of Model Prices .................. 95
6.1 Overview............................................... 95
6.2 ContractswithUnconditionalExerciseRights............... 96
6.3 ContractswithConditionalExerciseRights................. 97
6.3.1 CalculatingOptionPriceswiththeIFFT............. 97
6.3.2 Re?nement of the IFFT Pricing Algorithm . . . . . . . . . . . 101
6.3.3 Determination of the Optimal Parameters for the
NumericalScheme.................................103
7 Jump Speci?cations for A?ne Term-Structure Models . . . . . 111
7.1 Overview...............................................111
7.2 ExponentiallyDistributedJumps..........................115
7.3 NormallyDistributedJumps..............................117
7.4 GammaDistributedJumps ...............................120
8 Jump-Enhanced One-Factor Interest-Rate Models .........125
8.1 Overview...............................................125
8.2 TheOrnstein-UhlenbeckModel ...........................126

8.2.1 Derivation of the Characteristic Function . . . . . . . . . . . . . 126
8.2.2 NumericalResults.................................128
8.3 TheSquare-RootModel..................................136
8.3.1 Derivation of the Characteristic Function . . . . . . . . . . . . . 136
8.3.2 NumericalResults.................................138
9 Jump-Enhanced Two-Factor Interest-Rate Models .........145
9.1 Overview...............................................145
9.2 TheAdditiveOU-SRModel ..............................146
9.2.1 Derivation of the Characteristic Function . . . . . . . . . . . . . 146
9.2.2 NumericalResults.................................148
9.3 TheFong-VasicekModel .................................159
9.3.1 Derivation of the Characteristic Function . . . . . . . . . . . . . 159
9.3.2 NumericalResults.................................163
10 Non-A?ne Term-Structure Models and Short-Rate
Models with Stochastic Jump Intensity ....................171
10.1 Overview...............................................171
10.2 QuadraticGaussianModels...............................171
10.3 StochasticJumpIntensity ................................174
11 Conclusion ................................................175
A Derivation of the Complex-Valued Coe?cients for the
Characteristic Function in the Square-Root Model .........179
B Derivation of the Complex-Valued Coe?cients for the
Characteristic Function in the Fong-Vasicek Model ........183
References .....................................................187