The SABR LIBOR Market Model: Pricing, Calibration and Hedging for Complex Interest-Rate Derivatives

The SABR/LIBOR Market Model......Page 6
Contents......Page 8
Acknowledgements......Page 14
1 Introduction......Page 16
I The Theoretical Set-Up......Page 22
2 The LIBOR Market Model......Page 24
2.1 Definitions......Page 25
2.2 The Volatility Functions......Page 26
2.3 Separating the Correlation from the Volatility Term......Page 27
2.4 The Caplet-Pricing Condition Again......Page 29
2.5.1 The Simple Exponential Correlation......Page 31
2.5.2 The Multiplicative Correlation......Page 32
2.6 Possible Shapes of the Doust Correlation Function......Page 34
2.7 The Covariance Integral Again......Page 36
3.1 The SABR Model (and Why it is a Good Model)......Page 40
3.2 Description of the Model......Page 41
3.3 The Option Prices Given by the SABR Model......Page 42
3.4.2 The Normal Case (β = 0)......Page 43
3.5.1 Dependence on σ T \n0......Page 44
3.5.2 Dependence on β......Page 46
3.5.4 Dependence on ν......Page 48
3.6 The Link Between the Exponent, β, and the Volatility of Volatility, ν......Page 50
3.7 Volatility Clustering in the (LMM)-SABR Model......Page 52
3.8.1 Analysis of σ 0 (β = 0.5)......Page 55
3.8.2 Analysis of νT (β = 0.5)......Page 56
3.9 How Do We Know that the Market has Chosen β = 0.5?......Page 58
3.10.1 Log-Normality of the Volatility Process......Page 61
3.10.2 Problems with the (Stochastic) CEV Process......Page 62
4 The LMM-SABR Model......Page 66
4.1 The Equations of Motion......Page 67
4.2 The Nature of the Stochasticity Introduced by Our Model......Page 68
4.3 A Simple Correlation Structure......Page 69
4.4 A More General Correlation Structure......Page 70
4.5 Observations on the Correlation Structure......Page 72
4.6 The Volatility Structure......Page 73
4.8 The Volatility Structure in Periods of Market Stress......Page 74
4.9 A More General Stochastic Volatility Dynamics......Page 78
4.10.1 Preliminaries......Page 79
4.10.2 Standard LIBOR and LIBOR in Arrearsy......Page 85
4.10.3 LIBOR in Arrears: The Volatility Drift......Page 88
4.10.4 The Drifts in the General Case of Several Forward Ratesy......Page 89
4.10.5 Volatility Drifts in the Swap Measure......Page 90
II Implementation and Calibration......Page 94
5.1 The Caplet-Calibration Problem......Page 96
5.2 Choosing the Parameters of the Function, g(·), and the Initial Values, kT 0......Page 98
5.3 Choosing the Parameters of the Function h(·)......Page 99
5.5 Results......Page 103
5.6.1 Looking at Caplets in Isolation......Page 106
5.6.2 Looking at Caplets and Swaptions Together......Page 110
5.7 Implications for Model Choice......Page 114
6.1 The Swaption Calibration Problem......Page 116
6.2 Swap Rate and Forward Rate Dynamics......Page 117
6.3 Approximating the Instantaneous Swap Rate Volatility, St......Page 119
6.4 Approximating the Initial Value of the Swap Rate Volatility,  0 (First Route)......Page 120
6.5 Approximating  0 (Second Route) and the Volatility of Volatility of the Swap Rate, V......Page 121
6.7 Approximating the Swap Rate Exponent, B......Page 123
6.8.1 Comparison between Approximated and Simulation Prices......Page 124
6.8.2 Comparison between Parameters from the Approximations and the Simulations......Page 132
6.10 Appendix: Derivation of Approximate Swap Rate Volatility......Page 133
6.11 Appendix: Derivation of Swap-Rate/Swap-Rate-Volatility Correlation, RSABR......Page 135
6.12 Appendix: Approximation of dSt/St......Page 137
7.1 Statement of the Problem......Page 140
7.2 Creating a Valid Model Matrix......Page 141
7.2.2 First Strategy, Stage 2: Analytic Optimization of ci......Page 143
7.2.3 Second Strategy: Optimizing over Angles......Page 144
7.3 A Case Study: Calibration Using the Hypersphere Method......Page 146
7.4 Which Method Should One Choose?......Page 152
7.5 Appendix......Page 153
III Empirical Evidence......Page 156
8.1 Statement of the Empirical Problem......Page 158
8.2 What Do We Know from the Literature?......Page 160
8.3 Data Description......Page 163
8.4 Distributional Analysis and Its Limitations......Page 165
8.5 What is the True Exponent β?......Page 168
8.6 Appendix: Some Analytic Results......Page 170
9 Estimating the Volatility of the Forward Rates......Page 174
9.1 Expiry Dependence of Volatility of Forward Rates......Page 175
9.2 Direct Estimation......Page 177
9.3 Looking at the Normality of the Residuals......Page 179
9.4 Maximum-Likelihood and Variations on the Theme......Page 186
9.5 Information About the Volatility from the Options Market......Page 190
9.6 Overall Conclusions......Page 193
10.1 What We are Trying to Do......Page 196
10.2 Some Results from Random Matrix Theory......Page 197
10.4.1 The Forward-Rate/Forward-Rate Correlation Matrix......Page 200
10.4.2 The Forward-Rate/Volatility Correlation Block......Page 202
10.5.1 The Forward-Rate/Forward-Rate Correlation Matrix......Page 203
10.6 What Does Random Matrix Theory Really Tell Us?......Page 205
10.7 Calibrating the Correlation Matrices......Page 206
10.7.2 Results......Page 207
10.8.1 Eigenvalues of the Correlation Blocks......Page 210
10.8.2 Eigenvalues of Differences in the Correlation Blocks......Page 211
10.8.3 Entropy Measures......Page 213
10.8.4 The Forward-Rate/Volatility Correlation Block......Page 217
IV Hedging......Page 218
11.1 Statement of the Problem......Page 220
11.2.1 In- and Out-of-Model Hedging......Page 221
11.2.2 Functional-Dependence Hedging......Page 222
11.3 Definitions......Page 225
11.4.1 Delta Hedging......Page 226
11.4.2 Vega Hedging......Page 228
11.5.1 Vanna and Volga......Page 229
11.6 Generalizing Functional-Dependence Hedging......Page 230
11.7 How Does the Model Know about Vanna and Volga?......Page 234
11.8 Choice of Hedging Instrument......Page 235
12.1 Delta Hedging in the SABR-(LMM) Model......Page 236
12.2 Vega Hedging in the SABR-(LMM) Model......Page 244
13.2 Notation......Page 246
13.2.1 Estimation of the Unobservable Volatility......Page 247
13.2.3 Tests of the Hedging Performance of the LMM-SABR Model......Page 248
13.3 Hedging Results for the SABR Model......Page 249
13.4 Hedging Results for the LMM-SABR Model......Page 258
13.5 Conclusions......Page 260
14.1 The Intuition Behind the Problem......Page 262
14.2 Hedging the Forward-Rate Block......Page 264
14.3 Hedging the Volatility-Rate Block......Page 266
14.4 Hedging the Forward-Rate/Volatility Block......Page 268
14.5 Final Considerations......Page 269
15.1 Statement of the Problem......Page 272
15.2 The Volatility Function......Page 274
15.3 The Case Study......Page 275
15.4.1 The Normal-to-Normal State Transition......Page 276
15.4.2 The Normal-to-Excited Transition......Page 278
15.4.3 Normal-to-Unknown Transition......Page 280
15.5 Results......Page 281
15.5.2 Hedging Results for the Normal-to-Excited Transition......Page 282
15.6 Are We Getting Something for Nothing?......Page 285
References......Page 286
Index......Page 290