Table of Contents
Foreword v
Preface ix
1 Introduction 1
2 Market Framework 9
2.1 Studied Quantities 10
2.1.1 Financial Assets 10
2.1.2 Portfolios 12
2.1.3 Distribution Parameters 19
2.2 The Question of Time 22
2.2.1 Choosing the Measure of Time 22
2.2.2 Choosing the Scale of Time 25
3 Statistical Description of Markets 31
3.1 Construction of a Representation 32
3.1.1 Role of the Statistical Description 32
3.1.2 Continuous or Discontinuous Representations 32
3.2 Normality Tests 34
3.2.1 The Pearson-Fisher Coefficients 35
3.2.2 Kolmogorov Test 37
3.3 Discontinuity Test 39
3.3.1 Definition of Estimators 39
3.3.2 Confidence Intervals 41
3.4 Continuity Test 45
3.4.1 Definition of the Estimators 45
3.4.2 Confidence Interval 46
3.5 Testing the Finiteness of the Activity 47
3.5.1 Construction of the Tests 48
3.5.2 Illustration 50
4 Lévy Processes 53
4.1 Definitions and Construction 54
4.1.1 The Characteristic Exponent 54
4.1.2 Infinitely Divisible Distributions 54
4.1.3 A Construction with Poisson Processes 55
4.2 The Lévy-Khintchine Formula 60
4.2.1 Form of the Characteristic Exponent 60
4.2.2 The Lévy Measure 62
4.3 The Moments of Lévy Processes of Finite Variation 67
4.3.1 Existence of the Moments 68
4.3.2 Calculating the Moments 69
5 Stable Distributions and Processes 77
5.1 Definitions and Properties 78
5.1.1 Definitions 78
5.1.2 Characteristic Function and Lévy Measure 81
5.1.3 Some Special Cases of Stable Distributions 90
5.1.4 Simulating Paths of Stable Processes 94
5.2 Stable Financial Models 100
5.2.1 With Pure Stable Distributions 100
5.2.2 With Tempered Stable Distributions 101
6 Laplace Distributions and Processes 105
6.1 The First Laplace Distribution 106
6.1.1 The Intuitive Approach 106
6.1.2 Representations of the Laplace Distribution 108
6.1.3 Laplace Motion 117
6.2 The Asymmetrization of the Laplace Distribution 129
6.2.1 Construction of the Asymmetrization 129
6.2.2 Laplace Processes 134
6.3 The Laplace Distribution as the Limit of Hyperbolic Distributions 136
6.3.1 Motivation for Hyperbolic Distributions 138
6.3.2 Construction of Hyperbolic Distributions 139
6.3.3 Hyperbolic Distributions as Mixture Distributions 143
7 The Time Change Framework 147
7.1 Time Changes 148
7.1.1 Historical Survey 148
7.1.2 A First Modeling Example 149
7.2 Subordinated Brownian Motions 155
7.2.1 The Mechanics of Subordination 155
7.2.2 Construction of a Time Change 158
7.2.3 Brownian Motion in Gamma Time 165
7.3 Time-Changed Laplace Process 173
7.3.1 Mean-Reverting Clock 174
7.3.2 The Laplace Process in ICIR Time 178
8 Tail Distributions 181
8.1 Largest Values Approach 181
8.1.1 The Laws of Maxima 182
8.1.2 The Maxima of Lévy Processes 190
8.2 Threshold Approach 194
8.2.1 The Law of Threshold Exceedances 194
8.2.2 Linearity of Means beyond Thresholds 198
8.3 Statistical Phenomenon Approach 202
8.3.1 Concentration of Results 202
8.3.2 Hierarchy of Large Values 216
8.4 Estimation of the Shape Parameter 220
8.4.1 A New Algorithm 221
8.4.2 Examples of Results 224
9 Risk Budgets 227
9.1 Risk Measures 228
9.1.1 Main Issues 228
9.1.2 Definition of the Main Risk Measures 230
9.1.3 VaR, TCE, and the Laws of Maximum 233
9.1.4 Notion of Model Risk 235
9.2 Computation of Risk Budgets 242
9.2.1 Numerical Method 242
9.2.2 Semi-Heavy Distribution Tails 247
9.2.3 Heavy Distribution Tails 250
10 The Psychology of Risk 253
10.1 Basic Principles of the Psychology of Risk 254
10.1.1 The Notion of Psychological Value 254
10.1.2 The Notion of Certainty Equivalent 255
10.2 The Measurement of Risk Aversion 256
10.2.1 Definitions of the Risk Premium 256
10.2.2 Decomposition of the Risk Premium 258
10.2.3 Illustration 264
10.3 Typology of Risk Aversion 267
10.3.1 Attitude with Respect to Financial Risk 268
10.3.2 The Family of HARA Functions 269
11 Monoperiodic Portfolio Choice 275
11.1 The Optimization Program 277
11.2 Optimizing with Two Moments 279
11.2.1 One Risky Asset 280
11.2.2 Several Risky Assets 282
11.3 Optimizing with Three Moments 284
11.3.1 One Risky Asset 284
11.3.2 Several Risky Assets 288
11.4 Optimizing with Four Moments 289
11.4.1 One Risky Asset 289
11.4.2 Several Risky Assets 292
11.5 Other Problems 294
11.5.1 Giving up Comoments 294
11.5.2 Perturbative Approach and Normalized Moments 296
Appendix: Dealing with Uncertainty 297
12 Dynamic Portfolio Choice 303
12.1 The Optimization Program 304
12.1.1 The Objective Function 304
12.1.2 Modeling Stock Fluctuations 308
12.2 Classic Approach 315
12.3 Optimization in the Presence of Jumps 319
12.3.1 Presentation of the Model 319
12.3.2 Illustration 322
Appendix: Dealing with Uncertainty 325
13 Conclusion 331
Appendix A Concentration vs Diversification 333
Bibliography 341
Index 349
