A Probability Metrics Approach to Financial Risk Measures [Hardcover]

• 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 non-specialists 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

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 Lp-metrics between distribution functions 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 spaces 29

2.6.2 The equivalence of the notions of p. (semi-)distance on P2 l3v