Svetlozar Rachev, etc – A Probability Metric Approach to Financial Risk Measures

Description

Description

A Probability Metrics Approach to Financial Risk Measures relates the sector of likelihood metrics and danger measures to each other and applies them to finance for the primary time.

• Helps to reply the query: which danger measure is finest for a given drawback?
• Finds new relations between present lessons of danger measures
• Describes purposes in finance and extends them the place attainable
• Presents the idea of likelihood metrics in a extra accessible type which might be applicable for non-specialists within the discipline
• Applications embrace optimum portfolio alternative, danger concept, and numerical strategies in finance
• Topics requiring extra mathematical rigor and element are included in technical appendices to chapters

Table of Contents

Preface xiii

About the Authors xv

1 Introduction 1

1.1 Probability Metrics 1

1.2 Applications in Finance 2

2 Probability Distances and Metrics 7

2.1 Introduction 9

2.2 Some Examples of Probability Metrics 9

2.2.1 Engineer’s metric 10

2.2.2 Uniform (or Kolmogorov) metric 10

2.2.Three Lévy metric 11

2.2.4 Kantorovich metric 14

2.2.5 Lp-metrics between distribution features 15

2.2.6 Ky Fan metrics 16

2.2.7 Lp-metric 17

2.3 Distance and Semidistance Spaces 19

2.4 Definitions of Probability Distances and Metrics 24

2.5 Summary 28

2.6 Technical Appendix 28

2.6.1 Universally measurable separable metric areas 29

2.6.2 The equivalence of the notions of p. (semi-)distance on P2 and on X 35

3 Choice below Uncertainty 40

3.1 Introduction 41

3.2 Expected Utility Theory 44

3.2.1 St Petersburg Paradox 44

3.2.2 The von Neumann–Morgenstern anticipated utility concept 46

3.2.3 Types of utility features 48

3.3 Stochastic Dominance 51

3.3.1 First-order stochastic dominance 52

3.3.2 Second-order stochastic dominance 53

3.3.3 Rothschild–Stiglitz stochastic dominance 55

3.3.4 Third-order stochastic dominance 56

3.3.5 Efficient units and the portfolio alternative drawback 58

3.3.6 Return versus payoff 59

3.4 Probability Metrics and Stochastic Dominance 63

3.5 Cumulative Prospect Theory 66

3.6 Summary 70

3.7 Technical Appendix 70

3.7.1 The axioms of alternative 71

3.7.2 Stochastic dominance relations of order n 72

3.7.3 Return versus payoff and stochastic dominance 74

3.7.4 Other stochastic dominance relations 76

4 A Classification of Probability Distances 83

4.1 Introduction 86

4.2 Primary Distances and Primary Metrics 86

4.3 Simple Distances and Metrics 90

4.4 Compound Distances and Moment Functions 99

4.5 Ideal Probability Metrics 105

4.5.1 Interpretation and examples of ultimate likelihood metrics 107

4.5.2 Conditions for boundedness of ultimate likelihood metrics 112

4.6 Summary 114

4.7 Technical Appendix 114

4.7.1 Examples of major distances 114

4.7.2 Examples of easy distances 118

4.7.3 Examples of compound distances 131

4.7.4 Examples of second features 135

5 Risk and Uncertainty 146

5.1 Introduction 147

5.2 Measures of Dispersion 150

5.2.1 Standard deviation 151

5.2.2 Mean absolute deviation 153

5.2.3 Semi-standard deviation 154

5.2.4 Axiomatic description 155

5.2.5 Deviation measures 156

5.3 Probability Metrics and Dispersion Measures 158

5.4 Measures of Risk 159

5.4.1 Value-at-risk 160

5.4.2 Computing portfolio VaR in apply 165

5.4.3 Back-testing of VaR 172

5.4.4 Coherent danger measures 175

5.5 Risk Measures and Dispersion Measures 179

5.6 Risk Measures and Stochastic Orders 181

5.7 Summary 182

5.8 Technical Appendix 183

5.8.1 Convex danger measures 183

5.8.2 Probability metrics and deviation measures 184

5.8.3 Deviation measures and likelihood quasi-metrics 187

6 Average Value-at-Risk 191

6.1 Introduction 192

6.2 Average Value-at-Risk 193

6.2.1 AVaR for steady distributions 200

6.Three AVaR Estimation from a Sample 204

6.4 Computing Portfolio AVaR in Practice 207

6.4.1 The multivariate regular assumption 207

6.4.2 The historic methodology 208

6.4.3 The hybrid methodology 208

6.4.4 The Monte Carlo methodology 209

6.4.5 Kernel strategies 211

6.5 Back-testing of AVaR 218

6.6 Spectral Risk Measures 220

6.7 Risk Measures and Probability Metrics 223

6.8 Risk Measures Based on Distortion Functionals 226

6.9 Summary 227

6.10 Technical Appendix 228

6.10.1 Characteristics of conditional loss distributions 228

6.10.2 Higher-order AVaR 232

6.10.3 The minimization system for AVaR 234

6.10.Four ETL vs AVaR 237

6.10.5 Kernel-based estimation of AVaR 242

6.10.6 Remarks on spectral danger measures 245

7 Computing AVaR by Monte Carlo 252

7.1 Introduction 253

7.2 An Illustration of Monte Carlo Variability 256

7.3 Asymptotic Distribution, Classical Conditions 259

7.4 Rate of Convergence to the Normal Distribution 262

7.4.1 The impact of tail thickness 263

7.4.2 The impact of tail truncation 268

7.4.3 Infinite variance distributions 271

7.5 Asymptotic Distribution, Heavy-tailed Returns 277

7.6 Rate of Convergence, Heavy-tailed Returns 283

7.6.1 Stable Paretian distributions 283

7.6.2 Student’s t distribution 286

7.7 On the Choice of a Distributional Model 290

7.7.1 Tail conduct and return frequency 290

7.7.2 Practical implications 295

7.8 Summary 297

7.9 Technical Appendix 298

7.9.1 Proof of the steady restrict outcome 298

8 Stochastic Dominance Revisited 304

8.1 Introduction 306

8.2 Metrization of Preference Relations 308

8.3 The Hausdorff Metric Structure 310

8.4 Examples 314

8.4.1 The L´evy quasi-semidistance and first-order stochastic dominance 315

8.4.2 Higher-order stochastic dominance 317

8.4.3 The H-quasi-semidistance 320

8.4.Four AVaR generated stochastic orders 322

8.4.5 Compound quasi-semidistances 324

8.5 Utility-type Representations 325

8.6 Almost Stochastic Orders and Degree of Violation 328

8.7 Summary 330

8.8 Technical Appendix 332

8.8.1 Preference relations and topology 332

8.8.2 Quasi-semidistances and desire relations 334

8.8.3 Construction of quasi-semidistances on lessons of traders 335

8.8.4 Investors with balanced views 338

8.8.5 Structural classification of likelihood distances 339

Index 357

 

Author Information

Svetlozar (Zari) T. Rachev is Chair-Professor in Statistics, Econometrics and Mathematical Finance on the University of Karlsruhe within the School of Economics and Business Engineering. He can be Professor Emeritus on the University of California, Santa Barbara within the Department of Statistics and Applied Probability. He has printed seven monographs, eight handbooks and special-edited volumes, and over 300 analysis articles. His not too long ago coauthored books printed by Wiley in mathematical finance and monetary econometrics embrace Fat-Tailed and Skewed Asset Return Distributions: Implications for Risk Management, Portfolio choice, and Option Pricing (2005), Operational Risk: A Guide to Basel II Capital Requirements, Models, and Analysis (2007), Financial Econometrics: From Basics to Advanced Modeling Techniques (2007), and Bayesian Methods in Finance (2008).  He is cofounder of Bravo Group, now FinAnalytica, specializing in  monetary risk-management software program, for which he serves as Chief Scientist.

Stoyan V. Stoyanov, Ph.D. is the Head of Quantitative Research at FinAnalytica specializing in monetary danger administration software program. He is writer and co-author of quite a few papers a few of which have not too long ago appeared inEconomics Letters, Journal of Banking and Finance, Applied Mathematical Finance, Applied Financial Economics, andInternational Journal of Theoretical and Applied Finance. He is a coauthor of the mathematical finance e book Advanced Stochastic Models, Risk Assessment and Portfolio Optimization: the Ideal Risk, Uncertainty and Performance Measures(2008) printed by Wiley. Dr. Stoyanov has years of expertise in making use of optimum portfolio concept and market danger estimation strategies when fixing sensible issues of purchasers of FinAnalytica.

Frank J. Fabozzi is Professor within the Practice of Finance within the School of Management at Yale University. Prior to becoming a member of the Yale school, he was a Visiting Professor of Finance within the Sloan School at MIT. Professor Fabozzi is a Fellow of the International Center for Finance at Yale University and on the Advisory Council for the Department of Operations Research and Financial Engineering at Princeton University. He is the editor of the Journal of Portfolio Management. His not too long ago coauthored books printed by Wiley in mathematical finance and monetary econometrics embrace The Mathematics of Financial Modeling and Investment  Management (2004), Financial Modeling of the Equity Market: From CAPM to Cointegration (2006), Robust Portfolio Optimization and Management (2007), Financial Econometrics: From Basics to Advanced Modeling Techniques (2007), and Bayesian Methods in Finance (2008).

 

Reviews

 

The authors needs to be applauded for offering a novel and really readable account of likelihood metrics and the appliance of this specialised discipline to monetary issues.

Professor Carol Alexander, Henley Business School at Reading

This self-contained e book protecting the necessary discipline of likelihood metrics is a superb addition to the literature in monetary economics. What makes it distinctive is that it presents this space at a stage accessible to these with out in depth prior experience-academic and practitioner alike.

Petter Kolm, New York University