LIBOR Market Model in Practice

PRZEMYSLAW BACHERT is a senior financial engineer in the Global Financial Services Risk Management Group at Ernst and Young. He holds his Ph.D. in economics from the University of Lodz. In his work Przemyslaw is responsible for structure derivatives valuation and implementation of risk management systems. He has spent the last six years working with financial institutions in the Europe and Middle East to enhance their risk management capabilities including Algorithmics parameterization. Prior to joining Ernst and Young, Przemyslaw was a financial analyst at Bank Handlowy in Warsaw (Citigroup) where he was responsible for quantitative maintenance of front office system Kondor+. He is also a teacher in the Ernst and Young Academy of Business for the Financial Engineering course which covers the LIBOR Market Model. DARIUSZ GATAREK is Credit Risk Analyst at Glencore UK Ltd. In addition he is a professor at the WSB-National Louis University and the Polish Academy of Sciences. He joined Glencore UK Ltd from NumeriX LLC, where he was Director of Research specializing in interest rate derivatives pricing. Before he was involved in valuing derivatives and designing risk management systems for capital adequacy within the consultancy Deloitte and Touche and several banks. Dariusz has published a number of papers on financial models of which perhaps his work with Alan Brace and Marek Musiela on Brace-Gatarek-Musiela (BGM) models of interest rates dynamics is the most well-known. He is a frequent speaker at conferences worldwide.ROBERT MAKSYMIUK is a senior financial engineer in the Global Financial Services Risk Management Group at Ernst and Young where he is responsible for structured derivatives pricing and implementation of risk management systems for the clients. As consultant he has worked for several financial institutions in the Europe and Middle – East and his activity covered implementation Algo Suite risk management system. Prior to joining Ernst and Young Robert work in BRE Bank where he worked together with Dariusz Gatarek and he was engaged in quantitative research. Additionaly Robert is a teacher in the Ernst and Young Academy of Business for the Financial Engineering course which covers the LIBOR Market Model.

"The real contribution of the book to the existing literature is the hands-on description of the calibration algorithms." (Financial Markets Portfolio Management, 2007)

• Acknowledgments ixAbout the Authors xiIntroduction xiiiPart I THEORY 11 Mathematics in a Pill 31.1 Probability Space and Random Variables 31.2 Normal Distributions 41.3 Stochastic Processes 41.4 Wiener Processes 51.5 Geometric Wiener Processes 51.6 Markov Processes 61.7 Stochastic Integrals and Stochastic Differential Equations 61.8 Ito’s Formula 71.9 Martingales 71.10 Girsanov’s Theorem 71.11 Black’s Formula (1976) 81.12 Pricing Derivatives and Changing of Numeraire 81.13 Pricing of Interest Rate Derivatives and the Forward Measure 92 Heath-Jarrow-Morton and Brace-Gatarek-Musiela Models 132.1 HJM and BGM Models Under the Spot Measure 132.2 Vasi¡cek Model 162.3 Cox-Ingersoll-Ross Model 172.4 Black-Karasi´nski Model 172.5 HJM and BGM Models under the Forward Measures 183 Simulation 213.1 Simulation of HJM and BGM Models under the Forward Measure 213.2 Monte Carlo Simulation of Multidimensional Gaussian Variables 223.3 Trinomial Tree Simulation of Multidimensional Gaussian Variables 254 Swaption Pricing and Calibration 274.1 Linear Pricing in the BGM Model 294.2 Linear Pricing of Swaptions in the HJM Model 304.3 Universal Volatility Function 314.4 Time Homogeneous Volatility 334.5 Separated Volatility 344.6 Parametrized Volatility 374.7 Parametric Calibration to Caps and Swaptions Based on Rebonato Approach 384.8 Semilinear Pricing of Swaptions in the BGM Model 404.9 Semilinear Pricing of Swaptions in the HJM Model 414.10 Nonlinear Pricing of Swaptions 434.11 Examples 435 Smile Modelling in the BGM Model 455.1 The Shifted BGM Model 465.2 Stochastic Volatility for Long Term Options 485.3 The Uncertain Volatility Displaced LIBOR Market Model 505.4 Mixing the BGM and HJM Models 526 Simplified BGM and HJM Models 556.1 CMS Rate Dynamics in Single-Factor HJM Model 556.2 CMS Rate Dynamics in a Single Factor BGM Model 576.3 Calibration 586.4 Smile 59Part II CALIBRATION 637 Calibration Algorithms to Caps and Floors 677.1 Introduction 677.2 Market Data 677.3 Calibration to Caps 707.4 Non-Parametric Calibration Algorithms 787.5 Conclusions 868 Non-Parametric Calibration Algorithms to Caps and Swaptions 898.1 Introduction 898.2 The Separated Approach 908.3 The Separated Approach with Optimization 1098.4 The Locally Single Factor Approach 1178.5 Calibration with Historical Correlations of Forward Rates 1208.6 Calibration to Co-Terminal Swaptions 1258.7 Conclusions 1299 Calibration Algorithms to Caps and Swaptions Based on Optimization Techniques 1319.1 Introduction 1319.2 Non Parametric Calibration to Caps and Swaptions 1329.3 Parametric Method of Calibration 1579.4 Conclusions 166Part III SIMULATION 16710 Approximations of the BGM Model 17110.1 Euler Approximation 17110.2 Predictor-Corrector Approximation 17110.3 Brownian Bridge Approximation 17210.4 Combined Predictor-Corrector-Brownian Bridge 17310.5 Single-Dimensional Case 17410.6 Single-Dimensional Complete Case 17510.7 Binomial Tree Construction for LAn(t) 17710.8 Binomial Tree Construction for LDN(t) 18010.9 Numerical Example of Binomial Tree Construction 18110.10 Trinomial Tree Construction for LAN(t) 18810.11 Trinomial Tree Construction for LDN(t) 19110.12 Numerical Results 19210.13 Approximation of Annuities 19210.14 Swaption Pricing 19510.15 Lognormal Approximation 19810.16 Comparison 20010.17 Practical Example – Calibration to Co-terminal Swaptions and Simulation 20011 The One Factor LIBOR Markov Functional Model 20511.1 LIBOR Markov Functional Model Construction 20511.2 Binomial Tree Construction – Approach 1 20711.3 Binomial Tree Construction – Approach 2 21512 Optimal Stopping and Pricing of Bermudan Options 21912.1 Tree/Lattice Pricing 22012.2 Stochastic Meshes 22112.3 The Direct Method 22112.4 The Longstaff-Schwartz Method 22212.5 Additive Noise 22412.6 Example of BGM Dynamics 22812.7 Comparison of Methods 22813 Using the LSM Approach for Derivatives Valuation 22913.1 Pricing Algorithms 22913.2 Numerical Examples of Algorithms 13.1–13.4 23413.3 Calculation Results 25213.4 Some Theoretical Remarks on Optimal Stopping Under LSM 25313.5 Summary 257References 259Index 267