Title Details

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.

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