Extreme Financial Risks And Asset Allocation

by Olivier Le Courtois, Christian Walter

Overview

Each financial crisis calls for — by its novelty and the mechanisms it shares with preceding crises — appropriate means to analyze financial risks. In Extreme Financial Risks and Asset Allocation, the authors present in an accessible and timely manner the concepts, methods, and techniques that are essential for an understanding of these risks in an environment where asset prices are subject to sudden, rough, and unpredictable changes. These phenomena, mathematically known as “jumps”, play an important role in practice. Their quantitative treatment is generally tricky and is sparsely tackled in similar books. One of the main appeals of this book lies in its approachable and concise presentation of the ad hoc mathematical tools without sacrificing the necessary rigor and precision.

This book contains theories and methods which are usually found in highly technical mathematics books or in scattered, often very recent, research articles. It is a remarkable pedagogical work that makes these difficult results accessible to a large readership. Researchers, Masters and PhD students, and financial engineers alike will find this book highly useful.

Contents:

• Introduction
• Market Framework
• Statistical Description of Markets
• Lévy Processes
• Stable Distributions and Processes
• Laplace Distributions and Processes
• The Time Change Framework
• Tail Distributions
• Risk Budgets
• The Psychology of Risk
• Monoperiodic Portfolio Choice
• Dynamic Portfolio Choice
• Conclusion

Readership: Researchers, graduate students and financial engineers in the field of mathematical and quantitative finance.
Key Features:

• This book offers an excellent synthesis of the academic literature in a clear, ordered, and intuitive way
• The continuous-time theory of the choice of portfolio is exposed with particular care when asset dynamics are modeled with processes admitting a jump component. This is a technically difficult topic that is tackled here with a lot of clarity
• The collated works in this book facilitates access to the most recent techniques, making it user-friendly for readers

Product Details

ISBN-13:	9781783263103
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	01/21/2014
Series:	Series In Quantitative Finance , #5
Sold by:	Barnes & Noble
Format:	eBook
Pages:	372
File size:	14 MB
Note:	This product may take a few minutes to download.

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