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1 Introduction ................................ 1
2 Interest Rates and Related Contracts .................. 5
3 Estimating the Term-Structure ..................... 29
4 Arbitrage Theory ............................. 59
5 Short-Rate Models ............................ 79
6 Heath–Jarrow–Morton (HJM) Methodology .............. 93
6.1 ForwardCurveMovements ..................... 93
6.2 Absence of Arbitrage ........................ 95
6.3 Short-Rate Dynamics ........................ 96
6.4 HJM Models ............................. 97
6.4.1 Proportional Volatility . ................... 98
6.5 Fubini’s Theorem .......................... 99
6.6 Exercises...............................102
6.7 Notes.................................103
7 Forward Measures ............................105
7.1 T -BondasNumeraire ........................105
7.2 BondOptionPricing.........................109
7.2.1 Example: Vasiˇ cek Short-Rate Model . . . .........110
7.3 Black–Scholes Model with Gaussian Interest Rates .........110
7.3.1 Example: Black–Scholes–Vasiˇ cek Model . .........113
7.4 Exercises...............................114
7.5 Notes.................................116
8 Forwards and Futures ..........................117
8.1 ForwardContracts..........................117
8.2 FuturesContracts ..........................118
8.2.1 InterestRateFutures.....................119
8.3 Forwardvs.FuturesinaGaussianSetup ..............120
8.4 Exercises...............................121
8.5 Notes.................................122
9 Consistent Term-Structure Parametrizations .............123
9.1 Multi-factor Models .........................123
9.2 Consistency Condition ........................125
9.3 AffineTerm-Structures .......................127
9.4 Polynomial Term-Structures . . ...................128
9.4.1 Special Case: m = 1.....................129
9.4.2 General Case: m ≥ 1.....................131
9.5 Exponential–Polynomial Families ..................134
9.5.1 Nelson–Siegel Family . ...................134
9.5.2 SvenssonFamily.......................135
9.6 Exercises...............................138
9.7 Notes.................................140
10 Affine Processes ..............................143
10.1 Definition and Characterization of Affine Processes ........143
10.2 Canonical State Space ........................146
10.3 Discounting and Pricing in Affine Models .............151
10.3.1 Examples of Fourier Decompositions . . . .........157
10.3.2 Bond Option Pricing in Affine Models . . .........161xii Contents
10.3.3 Heston Stochastic Volatility Model .............166
10.4 Affine Transformations and Canonical Representation .......168
10.5 Existence and Uniqueness of Affine Processes . . .........171
10.6 On the Regularity of Characteristic Functions . . . .........173
10.7 Auxiliary Results for Differential Equations . . . .........177
10.7.1 Some Invariance Results ...................177
10.7.2 Some Results on Riccati Equations .............180
10.7.3 Proof of Theorem 10.3 ....................185
10.8Exercises...............................186
10.9Notes.................................194
11 Market Models ..............................197
11.1 HeuristicDerivation ........................197
11.2 LIBOR Market Model .......................199
11.2.1 LIBOR Dynamics Under Different Measures .......201
11.3 ImpliedBondMarket........................201
11.4 Implied Money-Market Account ..................204
11.5 SwaptionPricing..........................206
11.5.1ForwardSwapMeasure ..................207
11.5.2AnalyticApproximations..................209
11.6 Monte Carlo Simulation of the LIBOR Market Model ......210
11.7 Volatility Structure and Calibration ................212
11.7.1 Principal Component Analysis ...............212
11.7.2CalibrationtoMarketQuotes ...............213
11.8 Continuous-Tenor Case .......................219
11.9 Exercises ..............................221
11.10Notes ................................223
12 Default Risk ................................225
12.1 Default and Transition Probabilities .................225
12.2 Structural Approach .........................227
12.3 Intensity-Based Approach . . . ...................229
12.3.1 Construction of Doubly Stochastic Intensity-Based Models 235
12.3.2 Computation of Default Probabilities . . . .........236
12.3.3PricingDefaultRisk.....................236
12.3.4 Measure Change .......................240
12.4Exercises...............................242
12.5Notes.................................243
References ...................................245
Index ......................................253

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