Contents -- Foreword -- Preface -- 1. Introduction -- 2. Market Framework -- 2.1 Studied Quantities -- 2.1.1 Financial Assets -- 2.1.2 Portfolios -- 2.1.2.1 Investment Strategy -- 2.1.2.2 Sources of Performance -- 2.1.3 Distribution Parameters -- 2.1.3.1 Moments and Pearson-Fisher Coefficients -- 2.1.3.2 The Natural Computation Space -- 2.2 The Question of Time -- 2.2.1 Choosing the Measure of Time -- 2.2.2 Choosing the Scale of Time -- 2.2.2.1 The Change of Scale Rule -- 2.2.2.2 The Hypotheses for the Change of Scale Rule -- 3. Statistical Description of Markets -- 3.1 Construction of a Representation -- 3.1.1 Role of the Statistical Description -- 3.1.2 Continuous or Discontinuous Representations -- 3.1.2.1 The Debate on Continuity -- 3.1.2.2 Related Tests -- 3.2 Normality Tests -- 3.2.1 The Pearson-Fisher Coefficients -- 3.2.1.1 Asymmetry and Kurtosis Tests -- 3.2.1.2 Jarque-Bera Test -- 3.2.2 Kolmogorov Test -- 3.3 Discontinuity Test -- 3.3.1 Definition of Estimators -- 3.3.2 Confidence Intervals -- 3.3.2.1 Central Limit Theorem of Discontinuities -- 3.3.2.2 Illustration -- 3.4 Continuity Test -- 3.4.1 Definition of the Estimators -- 3.4.2 Confidence Interval -- 3.4.2.1 A New Indicator -- 3.4.2.2 Illustration -- 3.5 Testing the Finiteness of the Activity -- 3.5.1 Construction of the Tests -- 3.5.2 Illustration -- 4. Levy Processes -- 4.1 Definitions and Construction -- 4.1.1 The Characteristic Exponent -- 4.1.2 Infinitely Divisible Distributions -- 4.1.3 A Construction with Poisson Processes -- 4.1.3.1 The Simple Poisson Distribution -- 4.1.3.2 The Centered Poisson Distribution -- 4.1.3.3 The Poisson Process -- 4.1.3.4 The Compensated Poisson Process -- 4.1.3.5 The Compound Poisson Processes -- 4.1.3.6 The Compensated Compound Poisson Process -- 4.2 The Levy-Khintchine Formula -- 4.2.1 Form of the Characteristic Exponent.

4.2.2 The Levy Measure -- 4.2.2.1 The Activity of a Process -- 4.2.2.2 The Variation of a Process -- 4.2.2.3 Financial Models -- 4.3 The Moments of Levy Processes of Finite Variation -- 4.3.1 Existence of the Moments -- 4.3.2 Calculating the Moments -- 4.3.2.1 Calculation of the Mean (k = 1) -- 4.3.2.2 Calculation of the Variance (k = 2) -- 4.3.2.3 Calculation of the Skewness (k = 3) -- 4.3.2.4 Calculation of the Kurtosis (k = 4) -- 5. Stable Distributions and Processes -- 5.1 Definitions and Properties -- 5.1.1 Definitions -- 5.1.1.1 Stability Under Addition and Self-Similarity -- 5.1.1.2 Morphology of Alpha-Stable Distributions -- 5.1.2 Characteristic Function and Levy Measure -- 5.1.2.1 The Levy-Khintchine Representation -- 5.1.2.2 Integrated Form of the Characteristic Function -- 5.1.2.3 The Relationship Between the Two Parameterizations -- 5.1.2.4 Existence of Moments -- 5.1.3 Some Special Cases of Stable Distributions -- 5.1.3.1 1-stable or Cauchy Distribution -- 5.1.3.2 1/2-stable or Inverse Gaussian Distribution -- 5.1.3.3 Series Representations -- 5.1.4 Simulating Paths of Stable Processes -- 5.1.4.1 Construction Principle -- 5.1.4.2 Application to Stable Distributions -- 5.2 Stable Financial Models -- 5.2.1 With Pure Stable Distributions -- 5.2.1.1 The Mandelbrot Model -- 5.2.1.2 The Carr and Wu Model -- 5.2.2 With Tempered Stable Distributions -- 5.2.2.1 The Koponen Model -- 5.2.2.2 The CGMY Model -- 6. Laplace Distributions and Processes -- 6.1 The First Laplace Distribution -- 6.1.1 The Intuitive Approach -- 6.1.1.1 The Choice of a Law of Errors -- 6.1.1.2 The Supremacy of the Second Law of Errors -- 6.1.1.3 Discussion of the Construction Procedure -- 6.1.1.4 Two Different Universes of Thought -- 6.1.2 Representations of the Laplace Distribution -- 6.1.2.1 Density Functions -- 6.1.2.2 Characteristic Function.

6.1.2.4 Levy-Khintchine Formula -- 6.1.2.5 The Laplace Distribution as a Mixture Distribution -- 6.1.3 Laplace Motion -- 6.1.3.1 Definitions -- 6.1.3.2 Characteristic Function -- 6.1.3.3 Preliminary Results -- 6.1.3.4 Calculating the Characteristic Function -- 6.1.3.5 Laplace Motion: The Difference of Two Gamma Processes -- 6.1.3.6 The Sum of Laplace Distributions as a Mixture Distribution -- 6.2 The Asymmetrization of the Laplace Distribution -- 6.2.1 Construction of the Asymmetrization -- 6.2.1.1 The Principle of Asymmetrization -- 6.2.1.2 Characteristic Function -- 6.2.1.3 Trace of the Asymmetry -- 6.2.1.4 Infinitely Divisible Distribution and Levy Measure -- 6.2.2 Laplace Processes -- 6.2.2.1 Definitions -- 6.2.2.2 Sum of Generalized Laplace Distributions -- 6.2.2.3 The Laplace Process as Difference of two Gamma Processes -- 6.3 The Laplace Distribution as the Limit of Hyperbolic Distributions -- 6.3.1 Motivation for Hyperbolic Distributions -- 6.3.2 Construction of Hyperbolic Distributions -- 6.3.3 Hyperbolic Distributions as Mixture Distributions -- 7. The Time Change Framework -- 7.1 Time Changes -- 7.1.1 Historical Survey -- 7.1.2 A First Modeling Example -- 7.1.2.1 The Social Time of Exchanges -- 7.1.2.2 Market Capitalization in Social Time -- 7.1.2.3 Subordination of a Random Process -- 7.2 Subordinated Brownian Motions -- 7.2.1 The Mechanics of Subordination -- 7.2.1.1 Subordinators -- 7.2.1.2 Generalization -- 7.2.2 Construction of a Time Change -- 7.2.2.1 Subordinator and Characteristic Function -- 7.2.2.2 Non-Gaussianity in Calendar Time -- 7.2.3 Brownian Motion in Gamma Time -- 7.2.3.1 Choice of Gamma Time -- 7.2.3.2 Characteristic Function -- 7.2.3.3 Distribution Density -- 7.2.3.4 Difference of Two Gamma Processes -- 7.2.3.5 Levy Measure -- 7.2.3.6 Moments -- 7.3 Time-Changed Laplace Process -- 7.3.1 Mean-Reverting Clock.

7.3.1.1 Construction of an Auto-Regressive Process -- 7.3.1.2 Properties of the CIR Process -- 7.3.1.3 The Time Change Process -- 7.3.2 The Laplace Process in ICIR Time -- 8. Tail Distributions -- 8.1 Largest Values Approach -- 8.1.1 The Laws of Maxima -- 8.1.1.1 The Fisher-Tippett Theorem -- 8.1.1.2 The Generalized Jenkinson-von Mises Distribution -- 8.1.1.3 Maximum Domain of Attraction -- 8.1.2 The Maxima of Levy Processes -- 8.1.2.1 CGMY and Variance Gamma Processes -- 8.1.2.2 Alpha-Stable Motions -- 8.1.2.3 Hyperbolic Motions -- 8.1.2.4 Completely Asymmetric Alpha-Stable Distributions -- 8.2 Threshold Approach -- 8.2.1 The Law of Threshold Exceedances -- 8.2.1.1 Threshold Models -- 8.2.1.2 The Generalized Pareto Distribution -- 8.2.1.3 The Pareto Distribution in the Frechet Case -- 8.2.2 Linearity of Means beyond Thresholds -- 8.3 Statistical Phenomenon Approach -- 8.3.1 Concentration of Results -- 8.3.1.1 The Law of 80/20 -- 8.3.1.2 The Concentration Index and the Lorenz Curve -- 8.3.1.3 Application to a Generalized Pareto Distribution -- 8.3.1.4 Illustration -- 8.3.2 Hierarchy of Large Values -- 8.3.2.1 The Hierarchy Ratio -- 8.3.2.2 The Cumulative Frequency Curve -- 8.4 Estimation of the Shape Parameter -- 8.4.1 A New Algorithm -- 8.4.1.1 The Hill Method -- 8.4.1.2 An Algorithm: Automatic Estimation of the Threshold -- 8.4.2 Examples of Results -- 8.4.2.1 Theoretical Illustration -- 8.4.2.2 Application to the S&P 500 Index -- 9. Risk Budgets -- 9.1 Risk Measures -- 9.1.1 Main Issues -- 9.1.1.1 Coherent Risk Measures in Finance -- 9.1.1.2 Coherent Evaluation of Risk in Insurance -- 9.1.1.3 The Debate on Sub-Additivity -- 9.1.2 Definition of the Main Risk Measures -- 9.1.2.1 VaR -- 9.1.2.2 Mean Loss if VaR is Exceeded -- 9.1.2.3 Mean Excess Loss if VaR is Exceeded -- 9.1.2.4 Connection Between the Two Approaches.

9.1.3 VaR, TCE, and the Laws of Maximum -- 9.1.3.1 TCE as a Simple Multiple of VaR -- 9.1.3.2 Direct Computation with Pareto Distributions -- 9.1.4 Notion of Model Risk -- 9.1.4.1 Main Issues and Ideas -- 9.1.4.2 A First Illustration -- 9.1.4.3 Recurrence Time of a Large Fall of the S&P 500 Index -- 9.1.4.4 Computation of VaR on the USD/FRF Forex Rate -- 9.1.4.5 Computation of TCE with Two MDA Assumptions -- 9.2 Computation of Risk Budgets -- 9.2.1 Numerical Method -- 9.2.1.1 Computation Procedure -- 9.2.1.2 Computation of VaR -- 9.2.1.3 Computation of TCE -- 9.2.2 Semi-Heavy Distribution Tails -- 9.2.2.1 Risk Budgets with Laplace Processes -- 9.2.2.2 Risk Budgets with Laplace Processes in Deformed Time -- 9.2.3 Heavy Distribution Tails -- 9.2.3.1 VaR with an Alpha-Stable Motion -- 9.2.3.2 Model Risk and Factor 3 of the Basel Agreement -- 10. The Psychology of Risk -- 10.1 Basic Principles of the Psychology of Risk -- 10.1.1 The Notion of Psychological Value -- 10.1.2 The Notion of Certainty Equivalent -- 10.2 The Measurement of Risk Aversion -- 10.2.1 Definitions of the Risk Premium -- 10.2.1.1 The Risk Premium as a Difference in Evaluation -- 10.2.1.2 The Risk Premium as a Theoretical Selling Price -- 10.2.2 Decomposition of the Risk Premium -- 10.2.2.1 Approximation by Taylor Expansion -- 10.2.2.2 Definition of the Coefficients of the Psychology of Risk -- 10.2.2.3 Analysis of the Coefficients of the Psychology of Risk -- 10.2.3 Illustration -- 10.3 Typology of Risk Aversion -- 10.3.1 Attitude with Respect to Financial Risk -- 10.3.1.1 Risk Aversion Increasing with Wealth -- 10.3.1.2 Risk Aversion Decreasing with Wealth -- 10.3.1.3 Risk Aversion Constant with Respect to Wealth -- 10.3.2 The Family of HARA Functions -- 11. Monoperiodic Portfolio Choice -- 11.1 The Optimization Program -- 11.2 Optimizing with Two Moments -- 11.2.1 One Risky Asset.

11.2.2 Several Risky Assets.