SABR and SABR LIBOR Market Models in Practice With Examples Implemented in Python Christian Crispoldi Gerald Wigger Peter Larkin palgrave macmillan
Contents List of Figures ListofTables Acknowledgments List of Abbreviations List ofnotations xi xiv xvii xviii xix 1 Introduction 1 1.1 Who should read this book 1 1.2 Outline 2 1.3 Python, NumPy and SciPy 3 2 Interest Rate Derivatives Markets 5 2.1 Interest rates 5 2.2 What you need for trading: ISDAs, netting agreements and CSAs... 6 2.3 The evolution of complex derivatives trading 7 2.4 The effects of the financial credit crisis 8 3 Interest Rate Notions 11 3.1 Interest rate basics 11 3.2 The multiple curve framework 13 3.2.1 The OIS curve 15 3.2.2 The forward curve 17 3.2.3 Constructing tenor curves from tenor basis swaps 20 3.3 Interest rate valuations and measures 21 3.3.1 The spot measure 22 3.3.2 The terminal and forward measures 23 3.3.3 The swap measure 24 3.4 Volatility trading 24 3.4.1 Caps, floors - caplets, floorlets 25 3.4.2 Swaptions 26 4 Vanilla Models 29 4.1 Lognormal Black model 29 vii
viii f Contents 4.2 Normal model 34 4.3 Risk sensitivities 37 4.3.1 Delta 38 4.3.2 Gamma 59 4.3.3 Vega 4.3.4 Risk sensitivities computation 40 5 SABR Model 42 5.1 Introduction 42 5.2 SABR parameters 45 5.2.1 The at parameter 4 5 5.2.2 The ßk parameter 46 5.2.3 The v* parameter 47 5.2.4 The pk parameter 51 5.3 PDE and Kolmogorov equations 51 5.4 Hagan et al. approximations 55 5.4.1 Lognormal approximation 56 5.4.2 Normal approximation 59 5.5 SABR calibration in practice 60 5.5.1 Hagan et al. approximation calibration tests: fixed ßk 61 5.5.2 Hagan et al. approximation calibration tests: fixed pk 64 5.6 Risk sensitivities 67 5.6.1 SABR delta and SABR gamma 71 5.6.2 SABR vega 71 5.6.3 Smile skew and smile curvature sensitivities 76 5.7 Monte Carlo Simulation schemes for SABR 78 5.7.1 Monte Carlo Standard error 78 5.7.2 Pseudo random numbers 79 5.7.3 Generating correlated random numbers 79 5.7.4 Euler scheme 81 5.7.5 Milstein scheme 86 5.7.6 Antithetic sampling 91 5.8 The limits of Hagan et al. approximations 92 5.8.1 Risk neutral probability density Function implied by option prices 93 5.8.2 Risk neutral probability density function computation for the Hagan et al. approximations 93 5.8.3 Risk neutral probability density function tests for the Hagan et al. approximations 94 5.8.4 Accuracy tests for the Hagan et al. approximations 95 5.8.5 Explosive behavior for high strike options 98 5.9 Alternative SABR approximations 103
Contents f ix 5.9.1 Antonov et al. approximation, zero correlation case 105 5.9.2 Antonov et al. approximation, general correlation case 106 5.9.3 Antonov et al. approximation tests 110 5.10 Pricing in a negative forward rate regime: shifted SABR approximation 111 6 LI BOR Market Model 119 6.1 Introduction 119 6.1.1 Short rate models 119 6.1.2 Heath, Jarrow and Morton (HJM) framework 120 6.2 Dynamics of the LIBOR Market Model 125 6.2.1 Introduction 125 6.2.2 Lognormal dynamics under the terminal measure 125 6.3 The forward-forward correlation and its calibration 127 6.3.1 Historical forward-forward correlation estimation 128 6.3.2 Smoothing the historical forward-forward correlation: Svensson, Nelson and Siegel approach 130 6.3.3 Forward-forward correlation parametrization 132 6.3.3.1 Exponential parametrization 133 6.3.3.2 Exponential parametrization with decay control... 133 6.3.3.3 Double exponential parametrization 134 6.3.4 Forward-forward correlation calibration tests 135 6.4 Volatility parametrization and calibration 138 6.4.1 Calibrating to a term structure of ATM caplet volatilities... 138 6.4.2 Swap approximation in the LMM 141 6.4.3 Calibrating to the ATM swaption surface 142 6.5 Simulation 143 6.5.1 Factor reduction 143 6.5.2 Simulation under two-curve framework 148 6.5.3 Variance reduction techniques 149 6.5.3.1 Antithetic sampling 149 6.5.3.2 Control variates 150 6.5.3.3 Importance sampling 150 6.5.3.4 Moment matching 151 6.6 Risk sensitivities 152 6.6.1 Finite difference methods 152 6.6.2 Pathwise sensitivities and likelihood ratio methods 153 6.6.2.1 Glasserman and Zhao's pathwise sensitivities 153 6.6.3 Likelihood ratio methods 155 6.6.3.1 Likelihood ratio method and the LMM 158 6.6.4 Adjoint methods 160 6.6.4.1 Automatic differentiation 160
x f Contents 6.6.4.2 Giles-Glasserman method 164 6.6.4.3 Cost of calculations 165 6.6.4.4 LMMexample 166 7 SABR LIBOR Market Model 169 7.1 Introduction 169 7.2 Dynamics of the SABR LIBOR Market Model 170 7.2.1 Rebonato et al. drifts 171 7.2.2 Hagan and Lesniewski drifts 171 7.3 The correlation matrix FI and its calibration 172 7.3.1 Forward-forward correlation calibration 173 7.3.2 Volatility-volatility correlation calibration 173 7.3.2.1 Alternative correlation calibration methodology: correlations from GARCH volatilities 174 7.3.3 Volatility-volatility correlation calibration tests 175 7.3.4 Forward-volatility correlation calibration 175 7.3.4.1 Standard approach: diagonal parameters 179 7.3.4.2 Alternative approach: null foward-volatility correlations 180 7.4 Rebonato et al. SABR LMM parametrization 181 7.4.1 Volatility and volatility of volatility parametrizations 181 7.4.2 Diffusion processes 182 7.4.3 Calibratingg(Tk_i t) to a term structure of cq 183 7.4.4 Calibrating f) to a term structure of 185 7.4.5 Calibrating(f) 190 7.5 Simulation and pricing 193 7.5.1 Simulation of the forward rates under the terminal measure Q N - Python code 193 7.5.2 Pricing of forward dependent Instruments 197 7.5.2.1 Choosing the number oftime steps 197 7.5.2.2 Pricing of caplets and floorlets: numericai tests... 197 7.5.3 Pricing of swap dependent Instruments 198 7.5.3.1 Swap approximation using the SABR LMM Monte Carlo 198 7.5.3.2 Rebonato and White swaption approximation 199 A Appendices 203 A.l Time grid and day count Conventions 203 A.2 A note on hyperbolic geometry 206 A.3 LIBOR Market Model in the HJM framework 207 A.4 Swap Market Model 207 Bibliography 210 Index 214