Term-Structure Models (eBook)

Damir Filipovic is head of the Vienna Institute of Finance, a research institution in the field of Mathematical Finance, funded by the Vienna Science and Technology Fund (WWTF), and founded and co-funded by the University of Vienna and the Vienna University of Economics and Business Administration. Prior to this position he held the Chair of Financial and Insurance Mathematics at the University of Munich, and he was Assistant Professor at Princeton University. Moreover, he worked for the Swiss Federal Office of Private Insurance, where he co-developed the Swiss Solvency Test (SST) – a risk based solvency assessment for insurance undertakings – which was enacted in 2006. He also held visiting positions at ETH Zurich, Columbia University, Stanford University, and the Vienna University of Technology.

Preface 7
Contents 9
Introduction 13
Interest Rates and Related Contracts 16
Zero-Coupon Bonds 16
Interest Rates 17
Market Example: LIBOR 18
Simple vs. Continuous Compounding 19
Forward vs. Future Rates 20
Money-Market Account and Short Rates 20
Proxies for the Short Rate 21
Coupon Bonds, Swaps and Yields 22
Fixed Coupon Bonds 22
Floating Rate Notes 23
Interest Rate Swaps 23
Yield and Duration 26
Yield-to-Maturity 26
Duration and Convexity 27
Market Conventions 28
Day-Count Conventions 28
Coupon Bonds 29
Accrued Interest, Clean Price and Dirty Price 29
Yield-to-Maturity 30
Caps and Floors 30
Caps 31
Floors 31
Caps, Floors and Swaps 32
Black's Formula 32
Swaptions 33
Black's Formula 35
Exercises 35
Notes 38
Estimating the Term-Structure 40
A Bootstrapping Example 40
Non-parametric Estimation Methods 45
Bond Markets 46
Money Markets 47
Problems 49
Parametric Estimation Methods 49
Estimating the Discount Function with Cubic B-splines 49
Smoothing Splines 54
Exponential-Polynomial Families 60
Principal Component Analysis 62
Principal Components of a Random Vector 62
Sample Principle Components 63
PCA of the Forward Curve 64
Correlation 66
Exercises 67
Notes 68
Arbitrage Theory 69
Stochastic Calculus 69
Stochastic Integration 70
Quadratic Variation and Covariation 71
Itô's Formula 72
Stochastic Differential Equations 73
Stochastic Exponential 74
Financial Market 75
Self-Financing Portfolios 75
Numeraires 76
Arbitrage and Martingale Measures 77
Martingale Measures 78
Market Price of Risk 79
Admissible Strategies 80
The First Fundamental Theorem of Asset Pricing 80
Hedging and Pricing 81
Complete Markets 81
Arbitrage Pricing 84
Exercises 85
Notes 87
Short-Rate Models 88
Generalities 88
Diffusion Short-Rate Models 89
Examples 91
Inverting the Forward Curve 92
Affine Term-Structures 93
Some Standard Models 94
Vasicek Model 94
CIR Model 96
Dothan Model 97
Ho-Lee Model 98
Hull-White Model 99
Exercises 100
Notes 101
Heath-Jarrow-Morton (HJM) Methodology 102
Forward Curve Movements 102
Absence of Arbitrage 104
Short-Rate Dynamics 105
HJM Models 106
Proportional Volatility 107
Fubini's Theorem 108
Exercises 111
Notes 112
Forward Measures 113
T-Bond as Numeraire 113
Bond Option Pricing 117
Example: Vasicek Short-Rate Model 118
Black-Scholes Model with Gaussian Interest Rates 118
Example: Black-Scholes-Vasicek Model 121
Exercises 122
Notes 124
Forwards and Futures 125
Forward Contracts 125
Futures Contracts 126
Interest Rate Futures 127
Forward vs. Futures in a Gaussian Setup 128
Exercises 129
Notes 130
Consistent Term-Structure Parametrizations 131
Multi-factor Models 131
Consistency Condition 133
Affine Term-Structures 135
Polynomial Term-Structures 136
Special Case: m=1 137
General Case: m> =1
Exponential-Polynomial Families 142
Nelson-Siegel Family 142
Svensson Family 143
Exercises 146
Notes 148
Affine Processes 150
Definition and Characterization of Affine Processes 150
Canonical State Space 153
Discounting and Pricing in Affine Models 158
Examples of Fourier Decompositions 164
Bond Option Pricing in Affine Models 168
Example: Vasicek Short-Rate Model 169
Example: CIR Short-Rate Model 170
Heston Stochastic Volatility Model 173
Affine Transformations and Canonical Representation 175
Existence and Uniqueness of Affine Processes 178
On the Regularity of Characteristic Functions 180
Auxiliary Results for Differential Equations 184
Some Invariance Results 184
Some Results on Riccati Equations 187
Proof of Theorem 10.3 192
Exercises 193
Notes 201
Market Models 203
Heuristic Derivation 203
LIBOR Market Model 205
LIBOR Dynamics Under Different Measures 207
Implied Bond Market 207
Implied Money-Market Account 210
Swaption Pricing 212
Forward Swap Measure 213
Analytic Approximations 215
Monte Carlo Simulation of the LIBOR Market Model 216
Volatility Structure and Calibration 218
Principal Component Analysis 218
Calibration to Market Quotes 219
Continuous-Tenor Case 225
Exercises 227
Notes 229
Default Risk 230
Default and Transition Probabilities 230
Structural Approach 232
Intensity-Based Approach 234
Construction of Doubly Stochastic Intensity-Based Models 240
Computation of Default Probabilities 241
Pricing Default Risk 241
Zero Recovery 242
Partial Recovery at Maturity 243
Partial Recovery at Default 243
Measure Change 245
Exercises 247
Notes 248
References 249
Index 256