The LIBOR Market Model in Practice » książka
The LIBOR Market Model in Practice
ISBN-13: 9780470014431 / Angielski / Twarda / 2007 / 290 str.
The LIBOR Market Model in Practice
ISBN-13: 9780470014431 / Angielski / Twarda / 2007 / 290 str.
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The LIBOR market model is an interest rate model that is used to price derivatives. Whilst the number of books on interest rate modeling is large, this is a reflection of the speed of development of the theory and market demand.
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"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 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
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 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.
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