The LIBOR Market Model in Practice

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.

Autorentext

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.

Zusammenfassung
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.

Inhalt

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 …