Interest Rate Derivatives Explained: Volume 2 (eBook)

Jörg Kienitz is Partner at Quaternion Risk Management where he is responsible for business development, pricing models research and risk management consulting. Prior to this he was a Director at Deloitte and Co-lead of the quant team. Before joining Deloitte he was Head of Quantitative Analytics at Deutsche Postbank AG where he was involved in developing/implementing models for pricing, hedging and asset allocation. He lectures at university level on advanced financial modelling and implementation at the universities of Cape Town (UCT) and Wuppertal (BUW) where he is Adjunct Associate Professor, respectively Assistant Professor. Before that he lectured in the part time Masters programme at Oxford University on Financial Mathematics. He is a speaker at a number of major quant finance conferences including Global Derivatives and WBS Fixed Income. Jörg holds a PhD in Probability Theory from Bielefeld University.Peter Caspers is senior quantitative analyst at Quaternion Risk Management. He has over 17 years of experience as a quant in different banks and is a co-author of QuantLib, an open-source library for quantitative finance. He holds a degree in mathematics and computer science.

Contents 7
List of Figures 9
List of Tables 13
Goals of this Book and Global Overview 16
Introduction and Management Summary 16
Code 20
Further Reading 20
References 24
Products 27
1 Vanilla Bonds and Asset Swaps 28
1.1 Introduction and Objectives 28
1.2 The Z-Spread 28
1.3 Fixed Rate Bonds 29
1.4 Fixed Versus Float Vanilla Swaps 32
1.5 Hedging with Asset Swaps, the Credit Trap 36
1.6 Conclusion and Summary 37
References 38
2 Callability Features 39
2.1 Introduction and Objectives 39
2.2 Callability 39
2.3 Callable Fixed Bonds, Callable Swaps and Swaptions 42
2.4 Hedging and Model Calibration 46
2.5 Amortizing Structures 50
2.6 Callable Floater and Inverse Floater 53
2.7 Zero Bonds, Callable Zeros and Accreting Swaps 57
2.8 Conclusion and Summary 61
References 61
3 Structured Finance 62
3.1 Introduction and Objectives 62
3.2 Redemption Amount Variations 62
3.3 Draw Down Options 64
3.4 PIK Options and Capital Deferral Options 65
3.5 Tenor Options 66
3.6 Behavioural Models 67
3.7 Conclusion and Summary 67
4 More Exotic Features and Basis Risk Hedging 68
4.1 Introduction and Objectives 68
4.2 TaRNs and TaRN Swaps 68
4.3 Snowballs 71
4.4 Range Accruals 73
4.5 Volatility Notes 75
4.6 Structured Floating Rate Coupons 76
4.7 In-Currency Tenor Basis Swaps 77
4.8 Cross-Currency Basis Swaps 77
4.9 Conclusion and Summary 78
References 78
5 Exposures 79
5.1 Introduction and Objectives 79
5.2 Exposures---Summary and Illustrations 80
5.3 Examples 85
5.3.1 Interest Rate Swaps 86
5.3.2 Cross-Currency Swaps 87
5.3.3 Callable Swaps 90
5.4 Summary and Conclusions 92
Reference 92
Volatility 93
6 The Heston Model 94
6.1 Introduction and Objectives 94
6.2 Local Volatility Models 95
6.3 The Heston Model 98
6.4 Pricing 100
6.5 Applications and Calibration 103
6.5.1 Extensions of Heston 105
6.6 Conclusion and Summary 105
References 106
7 The SABR Model 107
7.1 Introduction and Objectives 107
7.2 Introduction 108
7.3 Approximation 113
7.4 Applications and Calibration 116
7.5 Numerical Technics for SABR 117
7.6 Extensions of SABR 126
7.6.1 Displaced/Normal SABR 126
7.6.2 ZABR 127
7.7 Recent Developments 133
7.7.1 Free SABR 133
7.7.2 Mixing SABR 137
7.8 Summary and Conclusions 139
References 140
Term Structure Models 142
8 Term Structure Models 143
8.1 Introduction and Objectives 143
8.2 Different Models for the Term Structure 143
8.2.1 Short Rate Models 145
8.2.2 Instantaneous Forward Rate Models 147
8.2.3 Market Models 147
8.3 Stochastic Volatility Enhancements 149
8.4 Stochastic Basis Spreads 149
8.4.1 Deterministic Basis 152
8.5 Pricing and Path Simulation 153
8.6 Summary and Conclusions 154
References 154
9 Short Rate Models 156
9.1 Introduction and Objectives 156
9.1.1 Gaussian Short Rate Model and the Hull--White Model 157
9.1.2 Affine Short Rate Models and the Cox--Ingersol--Ross Model (CIR) 174
9.2 Multi-dimensional Models/N-Factor Models 178
9.2.1 Example: The Two-Factor Gaussian Short Rate/G2++ Model 179
9.2.2 Short Rate Models for Hybrids 183
9.2.3 Dynamics of Zero-Coupon Bonds in a Gaussian Short Rate Model 183
9.2.4 Dynamics of HW-GBM FX Model Under the Domestic Risk-Neutral Measure QD 184
9.2.5 Dynamics of HW-GBM FX Model Under the Domestic T-forward Measure QDT 185
9.3 Stochastic Volatility 185
9.4 Stochastic Basis 186
9.5 Conclusion and Summary 189
References 189
10 A Gaussian Rates-Credit Pricing Framework 191
10.1 Introduction and Objectives 191
10.2 The Option-Adjusted Spread (OAS) 191
10.3 The 2F Rates-Credit LGM Model 192
10.4 Monte Carlo Paths in the 2F Rates-Credit LGM Model 195
10.5 Conclusion and Summary 197
References 197
11 Instantaneous Forward Rate Models and the Heath--Jarrow--Morton Framework 198
11.1 Introduction and Objectives 198
11.2 The Heath--Jarrow--Morton Framework 199
11.3 The Cheyette, Ritchken and Sankarasubramanian Model Class 200
11.3.1 The Hull--White Model 202
11.4 Cheyette Model Example 203
11.5 Summary and Conclusions 209
References 210
12 The Libor Market Model 211
12.1 Introduction and Objectives 211
12.2 Market Models 212
12.3 Libor Dynamics 212
12.3.1 Spot and Terminal Measure 214
12.3.2 Discretization 215
12.4 Modelling Volatility 217
12.5 Modelling Co-movement 218
12.5.1 Non-parametric 219
12.5.2 Parametric/Parsimonious 219
12.5.3 Factor Reduction 224
12.6 Interpolation 226
12.7 Libor Market and Swap Market Models 227
12.8 Extensions 229
12.9 Multi-tenor LMM and Stochastic Basis 231
12.10 Summary and Conclusions 232
References 233
A Numerical Techniques for Pricing and Exposure Modelling 234
A.1 Numerical Integration 234
A.1.1 Greeks 235
A.2 Fourier Transformation 235
A.2.1 Greeks 238
A.3 Finite Difference Techniques 238
A.3.1 Finite Differences 238
A.3.2 Finite Difference Schemes 239
A.3.2.1 Inner Scheme 239
A.3.2.2 Boundary Conditions 240
A.3.3 Consistency/Stability/Convergence 241
A.3.4 Solving for the Density 241
A.3.5 Solving for the Price 242
A.4 Monte Carlo Simulation 242
A.4.1 Random Numbers 244
A.4.1.1 Uniformly Distributed Randoms 244
A.4.1.2 Multiple Dimensions 244
A.4.1.3 Transforming to a Given Distribution 244
A.4.1.4 Multiple Dimensions 245
A.4.1.5 The Cholesky Decomposition 245
A.4.2 Path Simulation 246
A.4.2.1 The Exact Scheme 246
A.4.2.2 The Euler Scheme 247
A.4.2.3 The Predictor-Corrector Scheme 247
A.4.2.4 The Milstein Scheme 248
A.4.3 Averaging and Error Analysis 248
A.4.4 Special Purpose Schemes 251
A.5 The Longstaff-Schwartz Method 254
A.6 Further Considerations 256
References 256
Index 258