A Probability Metrics Approach to Financial Risk MeasuresISBN: 9781405183697
392 pages
February 2011, WileyBlackwell

Description
 Helps to answer the question: which risk measure is best for a given problem?
 Finds new relations between existing classes of risk measures
 Describes applications in finance and extends them where possible
 Presents the theory of probability metrics in a more accessible form which would be appropriate for nonspecialists in the field
 Applications include optimal portfolio choice, risk theory, and numerical methods in finance
 Topics requiring more mathematical rigor and detail 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
2.2.3 Lévy metric 11
2.2.4 Kantorovich metric 14
2.2.5 Lpmetrics between distribution functions 15
2.2.6 Ky Fan metrics 16
2.2.7 Lpmetric 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 spaces 29
2.6.2 The equivalence of the notions of p. (semi)distance on P2 and on X 35
3 Choice under 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 expected utility theory 46
3.2.3 Types of utility functions 48
3.3 Stochastic Dominance 51
3.3.1 Firstorder stochastic dominance 52
3.3.2 Secondorder stochastic dominance 53
3.3.3 Rothschild–Stiglitz stochastic dominance 55
3.3.4 Thirdorder stochastic dominance 56
3.3.5 Efficient sets and the portfolio choice problem 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 choice 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 ideal probability metrics 107
4.5.2 Conditions for boundedness of ideal probability metrics 112
4.6 Summary 114
4.7 Technical Appendix 114
4.7.1 Examples of primary distances 114
4.7.2 Examples of simple distances 118
4.7.3 Examples of compound distances 131
4.7.4 Examples of moment functions 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 Semistandard 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 Valueatrisk 160
5.4.2 Computing portfolio VaR in practice 165
5.4.3 Backtesting of VaR 172
5.4.4 Coherent risk 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 risk measures 183
5.8.2 Probability metrics and deviation measures 184
5.8.3 Deviation measures and probability quasimetrics 187
6 Average ValueatRisk 191
6.1 Introduction 192
6.2 Average ValueatRisk 193
6.2.1 AVaR for stable distributions 200
6.3 AVaR Estimation from a Sample 204
6.4 Computing Portfolio AVaR in Practice 207
6.4.1 The multivariate normal assumption 207
6.4.2 The historical method 208
6.4.3 The hybrid method 208
6.4.4 The Monte Carlo method 209
6.4.5 Kernel methods 211
6.5 Backtesting 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 Higherorder AVaR 232
6.10.3 The minimization formula for AVaR 234
6.10.4 ETL vs AVaR 237
6.10.5 Kernelbased estimation of AVaR 242
6.10.6 Remarks on spectral risk measures 245
7 Computing AVaR through 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 effect of tail thickness 263
7.4.2 The effect of tail truncation 268
7.4.3 Infinite variance distributions 271
7.5 Asymptotic Distribution, Heavytailed Returns 277
7.6 Rate of Convergence, Heavytailed 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 behavior 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 stable limit result 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 quasisemidistance and firstorder stochastic dominance 315
8.4.2 Higherorder stochastic dominance 317
8.4.3 The Hquasisemidistance 320
8.4.4 AVaR generated stochastic orders 322
8.4.5 Compound quasisemidistances 324
8.5 Utilitytype 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 Quasisemidistances and preference relations 334
8.8.3 Construction of quasisemidistances on classes of investors 335
8.8.4 Investors with balanced views 338
8.8.5 Structural classification of probability distances 339
Index 357
Author Information
Stoyan V. Stoyanov, Ph.D. is the Head of Quantitative Research at FinAnalytica specializing in financial risk management software. He is author and coauthor of numerous papers some of which have recently appeared in Economics Letters, Journal of Banking and Finance, Applied Mathematical Finance, Applied Financial Economics, and International Journal of Theoretical and Applied Finance. He is a coauthor of the mathematical finance book Advanced Stochastic Models, Risk Assessment and Portfolio Optimization: the Ideal Risk, Uncertainty and Performance Measures (2008) published by Wiley. Dr. Stoyanov has years of experience in applying optimal portfolio theory and market risk estimation methods when solving practical problems of clients of FinAnalytica.
Frank J. Fabozzi is Professor in the Practice of Finance in the School of Management at Yale University. Prior to joining the Yale faculty, he was a Visiting Professor of Finance in 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 recently coauthored books published by Wiley in mathematical finance and financial econometrics include 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 should be applauded for providing a unique and very
readable account of probability metrics and the application of this
specialized field to financial problems.
Professor Carol Alexander, Henley Business School at Reading
This selfcontained book covering the important field of probability metrics is a wonderful addition to the literature in financial economics. What makes it unique is that it presents this area at a level accessible to those without extensive prior experienceacademic and practitioner alike.
Petter Kolm, New York University
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