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

 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
About the Authors. Chapter 1 Introduction.
1.1 Probability Metrics.
1.2 Applications in Finance.
Chapter 2 Probability Distances and Metrics.
2.1 Introduction.
2.2 Some Examples of Probability Metrics.
2.2.1 Engineer's metric.
2.2.2 Uniform (or Kolmogorov) metric.
2.2.3 Levy metric.
2.2.4 Kantorovich metric.
2.2.5 Lpmetrics between distribution functions.
2.2.6 Ky Fan metrics.
2.2.7 Lpmetric.
2.3 Distance and Semidistance Spaces.
2.4 Definitions of Probability Distances and Metrics.
2.5 Summary.
2.6 Technical Appendix.
2.6.1 Universally measurable separable metric spaces.
2.6.2 The equivalence of the notions of p. (semi)distance on P2 and on X.
Chapter 3 Choice Under Uncertainty.
3.1 Introduction.
3.2 Expected Utility Theory.
3.2.1 St. Petersburg Paradox.
3.2.2 The von NeumannMorgenstern expected utility theory.
3.2.3 Types of utility functions.
3.3 Stochastic Dominance.
3.3.1 Firstorder stochastic dominance.
3.3.2 Secondorder stochastic dominance.
3.3.3 RothschildStiglitz stochastic dominance.
3.3.4 Thirdorder stochastic dominance.
3.3.5 Efficient sets and the portfolio choice problem.
3.3.6 Return versus payoff.
3.4 Probability Metrics and Stochastic Dominance.
3.5 Cumulative Prospect Theory.
3.6 Summary.
3.7 Technical Appendix.
3.7.1 The axioms of choice.
3.7.2 Stochastic dominance relations of order n.
3.7.3 Return versus payoff and stochastic dominance.
3.7.4 Other stochastic dominance relations.
Chapter 4 A Classification of Probability Distances.
4.1 Introduction.
4.2 Primary Distances and Primary Metrics.
4.3 Simple Distances and Metrics.
4.4 Compound Distances and Moment Functions.
4.5 Ideal Probability Metrics.
4.5.1 Interpretation and examples of ideal probability metrics.
4.5.2 Conditions for boundedness of ideal probability metrics.
4.6 Summary.
4.7 Technical Appendix.
4.7.1 Examples of primary distances.
4.7.2 Examples of simple distances.
4.7.3 Examples of compound distances.
4.7.4 Examples of moment functions.
Chapter 5 Risk and Uncertainty.
5.1 Introduction.
5.2 Measures of Dispersion.
5.2.1 Standard deviation.
5.2.2 Mean absolute deviation.
5.2.3 Semistandard deviation.
5.2.4 Axiomatic description.
5.2.5 Deviation measures.
5.3 Probability Metrics and Dispersion Measures.
5.4 Measures of Risk.
5.4.1 Valueatrisk.
5.4.2 Computing portfolio VaR in practice.
5.4.3 Backtesting of VaR.
5.4.4 Coherent risk measures.
5.5 Risk Measures and Dispersion Measures.
5.6 Risk Measures and Stochastic Orders.
5.7 Summary.
5.8 Technical Appendix.
5.8.1 Convex risk measures.
5.8.2 Probability metrics and deviation measures.
5.8.3 Deviation measures and probability quasimetrics.
Chapter 6 Average ValueatRisk.
6.1 Introduction.
6.2 Average ValueatRisk.
6.2.1 AVaR for stable distributions.
6.3 AVaR Estimation From a Sample.
6.4 Computing Portfolio AVaR in Practice.
6.4.1 The multivariate normal assumption.
6.4.2 The Historical Method.
6.4.3 The Hybrid Method.
6.4.4 The Monte Carlo Method.
6.4.5 Kernel methods.
6.5 Backtesting of AVaR.
6.6 Spectral Risk Measures.
6.7 Risk Measures and Probability Metrics.
6.8 Risk Measures Based on Distortion Functionals.
6.9 Summary.
6.10 Technical Appendix.
6.10.1 Characteristics of conditional loss distributions.
6.10.2 Higherorder AVaR.
6.10.3 The minimization formula for AVaR.
6.10.4 ETL vs AVaR.
6.10.5 Kernelbased estimation of AVaR.
6.10.6 Remarks on spectral risk measures.
Chapter 7 Computing AVaR through Monte Carlo.
7.1 Introduction.
7.2 An illustration of Monte Carlo Variability.
7.3 Asymptotic Distribution, Classical Conditions.
7.4 Rate of Convergence to the Normal Distribution.
7.4.1 The effect of tail thickness.
7.4.2 The effect of tail truncation.
7.4.3 Infinite variance distributions.
7.5 Asymptotic Distribution, Heavytailed Returns.
7.6 Rate of Convergence, Heavytailed Returns.
7.6.1 Stable Paretian distributions.
7.6.2 Student's t distribution.
7.7 On the choice of a distributional model.
7.7.1 Tail behavior and return frequency.
7.7.2 Practical implications.
7.8 Summary.
7.9 Technical Appendix.
7.9.1 Proof of the stable limit result.
Chapter 8 Stochastic Dominance Revisited.
8.1 Introduction.
8.2 Metrization of Preference Relations.
8.3 The Hausdorff Metric Structure.
8.4 Examples.
8.4.1 The Levy quasisemidistance and firstorder stochastic dominance.
8.4.2 Higher order stochastic dominance.
8.4.3 The Hquasisemidistance.
8.4.4 AVaR generated stochastic orders.
8.4.5 Compound quasisemidistances.
8.5 Utilitytype Representations.
8.6 Almost Sstochastic Orders and Degree of Violation.
8.7 Summary.
8.8 Technical Appendix.
8.8.1 Preference relations and topology.
8.8.2 Quasisemidistances and preference relations.
8.8.3 Construction of quasisemidistances on classes of investors.
8.8.4 Investors with balanced views.
8.8.5 Structural classification of probability distances.
Index.
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).
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