Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures

Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization
The finance industry is seeing increased interest in new risk measures and techniques for portfolio optimization when parameters of the model are uncertain.
This groundbreaking book extends traditional approaches of risk measurement and portfolio optimization by combining distributional models with risk or performance measures into one framework. Throughout these pages, the expert authors explain the fundamentals of probability metrics, outline new approaches to portfolio optimization, and discuss a variety of essential risk measures. Using numerous examples, they illustrate a range of applications to optimal portfolio choice and risk theory, as well as applications to the area of computational finance that may be useful to financial engineers. They also clearly show how stochastic models, risk assessment, and optimization are essential to mastering risk, uncertainty, and performance measurement.
Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization provides quantitative portfolio managers (including hedge fund managers), financial engineers, consultants, and?academic researchers with answers to the key question of which risk measure is best for any given problem.

Spis treści:
Preface.
Acknowledgments.
About the Authors.
CHAPTER 1: Concepts of Probability.
1.1 Introduction.
1.2 Basic Concepts.
1.3 Discrete Probability Distributions.
1.3.1 Bernoulli Distribution.
1.3.2 Binomial Distribution.
1.3.3 Poisson Distribution.
1.4 Continuous Probability Distributions.
1.4.1 Probability Distribution Function, Probability Density Function, and Cumulative Distribution Function.
1.4.2 The Normal Distribution.
1.4.3 Exponential Distribution.
1.4.4 Student’s t–distribution.
1.4.5 Extreme Value Distribution.
1.4.6 Generalized Extreme Value Distribution.
1.5 Stat istical Moments and Quantiles.
1.5.1 Location.
1.5.2 Dispersion.
1.5.3 Asymmetry.
1.5.4 Concentration in Tails.
1.5.5 Statistical Moments.
1.5.6 Quantiles.
1.5.7 Sample Moments.
1.6 Joint Probability Distributions.
1.6.1 Conditional Probability.
1.6.2 Definition of Joint Probability Distributions.
1.6.3 Marginal Distributions.
1.6.4 Dependence of Random Variables.
1.6.5 Covariance and Correlation.
1.6.6 Multivariate Normal Distribution.
1.6.7 Elliptical Distributions.
1.6.8 Copula Functions.
1.7 Probabilistic Inequalities.
1.7.1 Chebyshev’s Inequality.
1.7.2 Fr´echet–Hoeffding Inequality.
1.8 Summary.
CHAPTER 2: Optimization.
2.1 Introduction.
2.2 Unconstrained Optimization.
2.2.1 Minima and Maxima of a Differentiable Function.
2.2.2 Convex Functions.
2.2.3 Quasiconvex Functions.
2.3 Constrained Optimization.
2.3.1 Lagrange Multipliers.
2.3.2 Convex Programming.
2.3.3 Linear Programming.
2.3.4 Quadratic Programming.
2.4 Summary.
CHAPTER 3: Probability Metrics.
3.1 Introduction.
3.2 Measuring Distances: The Discrete Case.
3.2.1 Sets of Characteristics.
3.2.2 Distribution Functions.
3.2.3 Joint Distribution.
3.3 Primary, Simple, and Compound Metrics.
3.3.1 Axiomatic Construction.
3.3.2 Primary Metrics.
3.3.3 Simple Metrics.
3.3.4 Compound Metrics.
3.3.5 Minimal and Maximal Metrics.
3.4 Summary.
3.5 Technical Appendix.
3.5.1 Remarks on the Axiomatic Construction of Probability Metrics.
3.5.2 Examples of Probability Distances.
3.5.3 Minimal and Maximal Distances.
CHAPTER 4: Ideal Probability Metrics.
4.1 Introduction.
4.2 The Classical Central Limit Theorem.
4.2.1 The Binomial Approximation to the Normal Distribution.
4.2.2 The General Case.
4.2.3 Estimating the Distance from the Limit Distribution.
4.3 The Generalized Central Limit Theorem.
4.3.1 Stable Distribut ions.
4.3.2 Modeling Financial Assets with Stable Distributions.
4.4 Construction of Ideal Probability Metrics.
4.4.1 Definition.
4.4.2 Examples.
4.5 Summary.
4.6 Technical Appendix.
4.6.1 The CLT Conditions.
4.6.2 Remarks on Ideal Metrics.
CHAPTER 5: Choice under Uncertainty.
5.1 Introduction.
5.2 Expected Utility Theory.
5.2.1 St. Petersburg Paradox.
5.2.2 The von Neumann–Morgenstern Expected Utility Theory.
5.2.3 Types of Utility Functions.
5.3 Stochastic Dominance.
5.3.1 First–Order Stochastic Dominance.
5.3.2 Second–Order Stochastic Dominance.
5.3.3 Rothschild–Stiglitz Stochastic Dominance.
5.3.4 Third–Order Stochastic Dominance.
5.3.5 Efficient Sets and the Portfolio Choice Problem.
5.3.6 Return versus Payoff.
5.4 Probability Metrics and Stochastic Dominance.
5.5 Summary.
5.6 Technical Appendix.
5.6.1 The Axioms of Choice.
5.6.2 Stochastic Dominance Relations of Order n.
5.6.3 Return versus Payoff and Stochastic Dominance.
5.6.4 Other Stochastic Dominance Relations.
CHAPTER 6: Risk and Uncertainty.
6.1 Introduction.
6.2 Measures of Dispersion.
6.2.1 Standard Deviation.
6.2.2 Mean Absolute Deviation.
6.2.3 Semistandard Deviation.
6.2.4 Axiomatic Description.
6.2.5 Deviation Measures.
6.3 Probability Metrics and Dispersion Measures.
6.4 Measures of Risk.
6.4.1 Value–at–Risk.
6.4.2 Computing Portfolio VaR in Practice.
6.4.3 Backtesting of VaR.
6.4.4 Coherent Risk Measures.
6.5 Risk Measures and Dispersion Measures.
6.6 Risk Measures and Stochastic Orders.
6.7 Summary.
6.8 Technical Appendix.
6.8.1 Convex Risk Measures.
6.8.2 Probability Metrics and Deviation Measures.
CHAPTER 7: Average Value–at–Risk.
7.1 Introduction.
7.2 Average Value–at–Risk.
7.3 AVaR Estimation from a Sample.
7.4 Computing Portfolio AVaR in Pr actice.
7.4.1 The Multivariate Normal Assumption.
7.4.2 The Historical Method.
7.4.3 The Hybrid Method 217
7.4.4 The Monte Carlo Method.
7.5 Backtesting of AVaR.
7.6 Spectral Risk Measures.
7.7 Risk Measures and Probability Metrics.
7.8 Summary.
7.9 Technical Appendix.
7.9.1 Characteristics of Conditional Loss Distributions.
7.9.2 Higher–Order AVaR.
7.9.3 The Minimization Formula for AVaR.
7.9.4 AVaR for Stable Distributions.
7.9.5 ETL versus AVaR.
7.9.6 Remarks on Spectral Risk Measures.
CHAPTER 8: Optimal Portfolios.
8.1 Introduction.
8.2 Mean–Variance Analysis.
8.2.1 Mean–Variance Optimization Problems.
8.2.2 The Mean–Variance Efficient Frontier.
8.2.3 Mean–Variance Analysis and SSD.
8.2.4 Adding a Risk–Free Asset.
8.3 Mean–Risk Analysis.
8.3.1 Mean–Risk Optimization Problems.
8.3.2 The Mean–Risk Efficient Frontier.
8.3.3 Mean–Risk Analysis and SSD.
8.3.4 Risk versus Dispersion Measures.
8.4 Summary.
8.5 Technical Appendix.
8.5.1 Types of Constraints.
8.5.2 Quadratic Approximations to Utility Functions.
8.5.3 Solving Mean–Variance Problems in Practice.
8.5.4 Solving Mean–Risk Problems in Practice.
8.5.5 Reward–Risk Analysis.
CHAPTER 9: Benchmark Tracking Problems.
9.1 Introduction.
9.2 The Tracking Error Problem.
9.3 Relation to Probability Metrics.
9.4 Examples of r.d. Metrics.
9.5 Numerical Example.
9.6 Summary.
9.7 Technical Appendix.
9.7.1 Deviation Measures and r.d. Metrics.
9.7.2 Remarks on the Axioms.
9.7.3 Minimal r.d. Metrics.
CHAPTER 10: Performance Measures.
10.1 Introduction.
10.2 Reward–to–Risk Ratios.
10.2.1 RR Ratios and the Efficient Portfolios.
10.2.2 Limitations in the Application of Reward–to–Risk Ratios.
10.2.3 The STARR.
10.2.4 The Sortino Ratio.
10.2.5 The Sortino