ISBN: 3540421432
TITLE: The Measurement of Market Risk
AUTHOR: P.-Y. Moix
TOC:

1. Introduction 1
1.1 The Need for Risk Measurement 1
1.2 The Nature of Financial Risk 3
1.3 Formal Framework 5
1.3.1 Modelling the Uncertainty 6
1.3.2 The Information Structure 8
1.4 Problem Statement 9
1.5 Structure of the Book 11
1.6 Test, Environment 16
1.6.1 Environment I 17
1.6.2 Environment II 18
2. Risk and Risk Measures 21
2.1 The Investment Decision 22
2.1.1 Utility Theory and Expected Utility Hypothesis 23
2.1.2 Rules for the Ordering of Uncertain Prospects 33
2.2 The Capital Requirement Decision. 41
2.2.1 Value-at-Risk 42
2.2.2 Coherent Risk Measures 44
2.3 Summary 47
3. Modelling the Dynamics of the Risk Factors 49
3.1 Statistical Definitions 50
3.1.1 Stochastic Processes: Basic Definitions 50
3.1.2 Properties of Stochastic Processes 52
3.1.3 Basic Stochastic Processes 53
3.2 The Economic Assumption: the Efficient Market Hypothesis 59
3.3 Empirical Evidence for the Returns 62
3.3.1 Calendar Effects 62
3.3.2 Leptokurtosis and Weak Evidence of Skewness 62
3.3.3 The Autocorrelation of the Squared Returns 64
3.4 Models for the Risk Factor Dynamics 66
3.4.1 The Generic Model for the Log-returns 67
3.4.2 ARCH Models 68
3.4.3 Stochastic Variance Models 71
3.5 Empirical Analysis of the Returns on Swiss Stocks 80
3.5.1 The Data 80
3.5.2 Descriptive Statistics and Correlation 80
3.5.3 Implementation of an Alternative Model 82
3.5.4 Impact of the Alternative Modelling 90
3.6 Continuous-Time Models 92
3.7 Summary 97
4. Valuation of Financial Instruments 99
4.1 Principles of Valuation 100
4.1.1 Valuation by Arbitrage 102
4.2 Cash Instruments 116
4.2.1 Equities 116
4.2.2 Fixed-Income Instruments 117
4.3 Futures and Forwards 119
4.4 Options 121
4.4.1 The Black-Scholes Analysis 122
4.4.2 Risk-Neutral Valuation 124
4.4.3 Numerical Approaches 126
4.5 Approximation of the Value Function 133
4.5.1 Global Taylor Approximation for Option Pricing 135
4.5.2 Piecewise Taylor Approximations 136
5. Approximation of the Portfolio Distribution 141
5.1 Analytical Methods 142
5.1.1 Delta Approximation 144
5.1.2 Delta-Gamma Approximation 146
5.2 Generation of Scenarios 151
5.2.1 The Pseudo-Random Method 152
5.2.2 The Quasi-Random Method 153
5.2.3 Generation of Distributions for the Risk Factors 159
5.3 Monte Carlo Simulation 165
5.3.1 Error Analysis 166
5.3.2 Variance Reduction Techniques 173
5.4 The BDPQA 17 7
5.4.1 Simplices 178
5.4.2 Simplicial Coverage of the Risk Factor Distribution 180
5.4.3 Barycentric Discretisation 183
5.4.4 Approximation of the Portfolio Distribution 185
5.4.5 Refinement Strategies 188
5.4.6 Numerical Example 191
5.5 Benchmarking the BDPQX 195
5.5.1 The Choice of the Holding Period 201
5.6 Summary 202
6. Sample Estimation of Risk Measures 205
6.1 Introduction 205
6.2 Order Statistics 206
6.2.1 Distribution of Order Statistics 206
6.2.2 Moments of Order Statistics 209
6.2.3 Confidence Interval for Population Quantiles 210
6.3 Quantile Estimators Based on Order Statistics 213
6.3.1 Linear Combination of Several Order Statistics 214
6.4 Kernel-Based Estimators 215
6.4.1 Accuracy of the Estimate Density 217
6.4.2 Bandwidth Selection 221
6.4.3 Quantile Estimation Based on the Kernel Density Method 223
6.5 Comparison of the Quantile Estimators 224
6.6 Summary 227
7. Conclusion and Outlook 229
7.1 Summary 229
7.2 The Issue of Credit Risk 232
7.3 Outlook 233
A. Probability and Statistics 235
A.1 Probabilistic Modelling 235
A.2 Random Variable 237
A.2.1 Distribution Function 238
A.2.2 Moments 239
A.2.3 Independence and Correlation 241
4.2.4 Conditional Probability and Expectation 242
A.2.5 Stochastic Processes and Information Structure 243
A.2.6 Martingales 244
4.3 Selected Distributions 245
A.3.1 Basic Distributions 245
A.3.2 Elliptically Contoured Distributions 246
A.3.3 Stable Distribution 248
A.4 Types of Convergence 249
A.5 Sampling Theory 251
Bibliography 253
List of Figures 265
List of Tables 267
Index 269
END
