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

by Rachev, Svetlozar T.; Stoyanov, Stoyan V.; Fabozzi, Frank J.

• ISBN13: 9780470053164
• ISBN10: 047005316X
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-02-25
• Publisher: Wiley

Summary

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.

Author Biography

Svetlozar T. Rachev, PhD, Doctor of Science, is Chair-Professor at the University of Karlsruhe in the School of Economics and Business Engineering; Professor Emeritus at the University of California, Santa Barbara; and Chief-Scientist of FinAnalytica Inc.

Stoyan V. Stoyanov, PhD, is the Chief Financial Researcher at FinAnalytica Inc.

Frank J. Fabozzi, PhD, CFA, is Professor in the Practice of Finance and Becton Fellow at Yale University's School of Management and the Editor of the Journal of Portfolio Management.

Table of Contents

Preface

Acknowledgments About the Authors

Concepts of Probability

Introduction

Basic Concepts

Discrete Probability Distributions

Bernoulli Distribution

Binomial Distribution

Poisson Distribution

Continuous Probability Distributions

Probability Distribution Function, Probability Density Function, and Cumulative Distribution Function

The Normal Distribution

Exponential Distribution

Student's t-distribution

Extreme Value Distribution

Generalized Extreme Value Distribution

Statistical Moments and Quantiles

Location

Dispersion

Asymmetry

Concentration in Tails

Statistical Moments

Quantiles

Sample Moments

Joint Probability Distributions

Conditional Probability

Definition of Joint Probability Distributions

Marginal Distributions

Dependence of Random Variables

Covariance and Correlation

Multivariate Normal Distribution

Elliptical Distributions

Copula Functions

Probabilistic Inequalities

Chebyshev's Inequality

Fr'echet-Hoeffding Inequality

Summary

Optimization

Introduction

Unconstrained Optimization

Minima and Maxima of a Differentiable Function

Convex Functions

Quasiconvex Functions

Constrained Optimization

Lagrange Multipliers

Convex Programming

Linear Programming

Quadratic Programming

Summary

Probability Metrics

Introduction

Measuring Distances: The Discrete Case

Sets of Characteristics

Distribution Functions

Joint Distribution

Primary, Simple, and Compound Metrics

Axiomatic Construction

Primary Metrics

Simple Metrics

Compound Metrics

Minimal and Maximal Metrics

Summary

Technical Appendix

Remarks on the Axiomatic Construction of Probability Metrics

Examples of Probability Distances

Minimal and Maximal Distances

Ideal Probability Metrics

Introduction

The Classical Central Limit Theorem

The Binomial Approximation to the Normal Distribution

The General Case

Estimating the Distance from the Limit Distribution

The Generalized Central Limit Theorem

Stable Distributions

Modeling Financial Assets with Stable Distributions

Construction of Ideal Probability Metrics

Definition

Examples

Summary

Technical Appendix

The CLT Conditions

Remarks on Ideal Metrics

Choice under Uncertainty

Introduction

Expected Utility Theory

St. Petersburg Paradox

The von Neumann-Morgenstern Expected Utility Theory

Types of Utility Functions

Stochastic Dominance

First-Order Stochastic Dominance

Second-Order Stochastic Dominance

Rothschild-Stiglitz Stochastic Dominance

Third-Order Stochastic Dominance

Efficient Sets and the Portfolio Choice Problem

Return versus Payoff

Probability Metrics and Stochastic Dominance

Summary

Technical Appendix

The Axioms of Choice

Stochastic Dominance Relations of Order n

Return versus Payoff and Stochastic

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