With Examples Implemented in Python

Transcription

1 SABR and SABR LIBOR Market Models in Practice With Examples Implemented in Python Christian Crispoldi Gerald Wigger Peter Larkin palgrave macmillan

2 Contents List of Figures ListofTables Acknowledgments List of Abbreviations List ofnotations xi xiv xvii xviii xix 1 Introduction Who should read this book Outline Python, NumPy and SciPy 3 2 Interest Rate Derivatives Markets Interest rates What you need for trading: ISDAs, netting agreements and CSAs The evolution of complex derivatives trading The effects of the financial credit crisis 8 3 Interest Rate Notions Interest rate basics The multiple curve framework The OIS curve The forward curve Constructing tenor curves from tenor basis swaps Interest rate valuations and measures The spot measure The terminal and forward measures The swap measure Volatility trading Caps, floors - caplets, floorlets Swaptions 26 4 Vanilla Models Lognormal Black model 29 vii

3 viii f Contents 4.2 Normal model Risk sensitivities Delta Gamma Vega Risk sensitivities computation 40 5 SABR Model Introduction SABR parameters The at parameter The ßk parameter The v* parameter The pk parameter PDE and Kolmogorov equations Hagan et al. approximations Lognormal approximation Normal approximation SABR calibration in practice Hagan et al. approximation calibration tests: fixed ßk Hagan et al. approximation calibration tests: fixed pk Risk sensitivities SABR delta and SABR gamma SABR vega Smile skew and smile curvature sensitivities Monte Carlo Simulation schemes for SABR Monte Carlo Standard error Pseudo random numbers Generating correlated random numbers Euler scheme Milstein scheme Antithetic sampling The limits of Hagan et al. approximations Risk neutral probability density Function implied by option prices Risk neutral probability density function computation for the Hagan et al. approximations Risk neutral probability density function tests for the Hagan et al. approximations Accuracy tests for the Hagan et al. approximations Explosive behavior for high strike options Alternative SABR approximations 103

4 Contents f ix Antonov et al. approximation, zero correlation case Antonov et al. approximation, general correlation case Antonov et al. approximation tests Pricing in a negative forward rate regime: shifted SABR approximation LI BOR Market Model Introduction Short rate models Heath, Jarrow and Morton (HJM) framework Dynamics of the LIBOR Market Model Introduction Lognormal dynamics under the terminal measure The forward-forward correlation and its calibration Historical forward-forward correlation estimation Smoothing the historical forward-forward correlation: Svensson, Nelson and Siegel approach Forward-forward correlation parametrization Exponential parametrization Exponential parametrization with decay control Double exponential parametrization Forward-forward correlation calibration tests Volatility parametrization and calibration Calibrating to a term structure of ATM caplet volatilities Swap approximation in the LMM Calibrating to the ATM swaption surface Simulation Factor reduction Simulation under two-curve framework Variance reduction techniques Antithetic sampling Control variates Importance sampling Moment matching Risk sensitivities Finite difference methods Pathwise sensitivities and likelihood ratio methods Glasserman and Zhao's pathwise sensitivities Likelihood ratio methods Likelihood ratio method and the LMM Adjoint methods Automatic differentiation 160

5 x f Contents Giles-Glasserman method Cost of calculations LMMexample SABR LIBOR Market Model Introduction Dynamics of the SABR LIBOR Market Model Rebonato et al. drifts Hagan and Lesniewski drifts The correlation matrix FI and its calibration Forward-forward correlation calibration Volatility-volatility correlation calibration Alternative correlation calibration methodology: correlations from GARCH volatilities Volatility-volatility correlation calibration tests Forward-volatility correlation calibration Standard approach: diagonal parameters Alternative approach: null foward-volatility correlations Rebonato et al. SABR LMM parametrization Volatility and volatility of volatility parametrizations Diffusion processes Calibratingg(Tk_i t) to a term structure of cq Calibrating f) to a term structure of Calibrating(f) Simulation and pricing Simulation of the forward rates under the terminal measure Q N - Python code Pricing of forward dependent Instruments Choosing the number oftime steps Pricing of caplets and floorlets: numericai tests Pricing of swap dependent Instruments Swap approximation using the SABR LMM Monte Carlo 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