The LIBOR Market Model (LMM) is the first model of interest rates dynamics consistent with the market practice of pricing interest rate derivatives and therefore it is widely used by financial institution for valuation of interest rate derivatives. This book provides a full practitioner's approach to the LIBOR Market Model. It adopts the specific language of a quantitative analyst to the largest possible level and is one of first books on the subject written entirely by quants. The book is divided into three parts - theory, calibration and simulation. New and important issues are covered, such as various drift approximations, various parametric and nonparametric calibrations, and the uncertain volatility approach to smile modelling; a version of the HJM model based on market observables and the duality between BGM and HJM models. Co-authored by Dariusz Gatarek, the 'G' in the BGM model who is internationally known for his work on LIBOR market models, this book offers an essential perspective on the global benchmark for short-term interest rates.

Table of Contents:
Acknowledgments ix About the Authors xi Introduction xiii Part I THEORY 1 1 Mathematics in a Pill 3 1.1 Probability Space and Random Variables 3 1.2 Normal Distributions 4 1.3 Stochastic Processes 4 1.4 Wiener Processes 5 1.5 Geometric Wiener Processes 5 1.6 Markov Processes 6 1.7 Stochastic Integrals and Stochastic Differential Equations 6 1.8 Ito’s Formula 7 1.9 Martingales 7 1.10 Girsanov’s Theorem 7 1.11 Black’s Formula (1976) 8 1.12 Pricing Derivatives and Changing of Numeraire 8 1.13 Pricing of Interest Rate Derivatives and the Forward Measure 9 2 Heath-Jarrow-Morton and Brace-Gatarek-Musiela Models 13 2.1 HJM and BGM Models Under the Spot Measure 13 2.2 Vasi¡cek Model 16 2.3 Cox-Ingersoll-Ross Model 17 2.4 Black-Karasi´nski Model 17 2.5 HJM and BGM Models under the Forward Measures 18 3 Simulation 21 3.1 Simulation of HJM and BGM Models under the Forward Measure 21 3.2 Monte Carlo Simulation of Multidimensional Gaussian Variables 22 3.3 Trinomial Tree Simulation of Multidimensional Gaussian Variables 25 4 Swaption Pricing and Calibration 27 4.1 Linear Pricing in the BGM Model 29 4.2 Linear Pricing of Swaptions in the HJM Model 30 4.3 Universal Volatility Function 31 4.4 Time Homogeneous Volatility 33 4.5 Separated Volatility 34 4.6 Parametrized Volatility 37 4.7 Parametric Calibration to Caps and Swaptions Based on Rebonato Approach 38 4.8 Semilinear Pricing of Swaptions in the BGM Model 40 4.9 Semilinear Pricing of Swaptions in the HJM Model 41 4.10 Nonlinear Pricing of Swaptions 43 4.11 Examples 43 5 Smile Modelling in the BGM Model 45 5.1 The Shifted BGM Model 46 5.2 Stochastic Volatility for Long Term Options 48 5.3 The Uncertain Volatility Displaced LIBOR Market Model 50 5.4 Mixing the BGM and HJM Models 52 6 Simplified BGM and HJM Models 55 6.1 CMS Rate Dynamics in Single-Factor HJM Model 55 6.2 CMS Rate Dynamics in a Single Factor BGM Model 57 6.3 Calibration 58 6.4 Smile 59 Part II CALIBRATION 63 7 Calibration Algorithms to Caps and Floors 67 7.1 Introduction 67 7.2 Market Data 67 7.3 Calibration to Caps 70 7.4 Non-Parametric Calibration Algorithms 78 7.5 Conclusions 86 8 Non-Parametric Calibration Algorithms to Caps and Swaptions 89 8.1 Introduction 89 8.2 The Separated Approach 90 8.3 The Separated Approach with Optimization 109 8.4 The Locally Single Factor Approach 117 8.5 Calibration with Historical Correlations of Forward Rates 120 8.6 Calibration to Co-Terminal Swaptions 125 8.7 Conclusions 129 9 Calibration Algorithms to Caps and Swaptions Based on Optimization Techniques 131 9.1 Introduction 131 9.2 Non Parametric Calibration to Caps and Swaptions 132 9.3 Parametric Method of Calibration 157 9.4 Conclusions 166 Part III SIMULATION 167 10 Approximations of the BGM Model 171 10.1 Euler Approximation 171 10.2 Predictor-Corrector Approximation 171 10.3 Brownian Bridge Approximation 172 10.4 Combined Predictor-Corrector-Brownian Bridge 173 10.5 Single-Dimensional Case 174 10.6 Single-Dimensional Complete Case 175 10.7 Binomial Tree Construction for LAn(t) 177 10.8 Binomial Tree Construction for LDN(t) 180 10.9 Numerical Example of Binomial Tree Construction 181 10.10 Trinomial Tree Construction for LAN(t) 188 10.11 Trinomial Tree Construction for LDN(t) 191 10.12 Numerical Results 192 10.13 Approximation of Annuities 192 10.14 Swaption Pricing 195 10.15 Lognormal Approximation 198 10.16 Comparison 200 10.17 Practical Example – Calibration to Co-terminal Swaptions and Simulation 200 11 The One Factor LIBOR Markov Functional Model 205 11.1 LIBOR Markov Functional Model Construction 205 11.2 Binomial Tree Construction – Approach 1 207 11.3 Binomial Tree Construction – Approach 2 215 12 Optimal Stopping and Pricing of Bermudan Options 219 12.1 Tree/Lattice Pricing 220 12.2 Stochastic Meshes 221 12.3 The Direct Method 221 12.4 The Longstaff-Schwartz Method 222 12.5 Additive Noise 224 12.6 Example of BGM Dynamics 228 12.7 Comparison of Methods 228 13 Using the LSM Approach for Derivatives Valuation 229 13.1 Pricing Algorithms 229 13.2 Numerical Examples of Algorithms 13.1–13.4 234 13.3 Calculation Results 252 13.4 Some Theoretical Remarks on Optimal Stopping Under LSM 253 13.5 Summary 257 References 259 Index 267