Fat-Tailed and Skewed Asset Return Distributions Implications for Risk Management, Portfolio Selection, and Option Pricing

by Rachev, Svetlozar T.; Menn, Christian; Fabozzi, Frank J.

• ISBN13: 9780471718864
• ISBN10: 0471718866
• Edition: 1st
• Format: Hardcover
• Copyright: 2005-08-05
• Publisher: Wiley

Summary

While mainstream financial theories and applications assume that asset returns are normally distributed, overwhelming empirical evidence shows otherwise. Yet many professionals don't appreciate the highly statistical models that take this empirical evidence into consideration. Fat-Tailed and Skewed Asset Return Distributions examines this dilemma and offers readers a less technical look at how portfolio selection, risk management, and option pricing modeling should and can be undertaken when the assumption of a non-normal distribution for asset returns is violated. Topics covered in this comprehensive book include an extensive discussion of probability distributions, estimating probability distributions, portfolio selection, alternative risk measures, and much more. Fat-Tailed and Skewed Asset Return Distributions provides a bridge between the highly technical theory of statistical distributional analysis, stochastic processes, and econometrics of financial returns and real-world risk management and investments.

Author Biography

SVETLOZAR T. RACHEV, PhD, DR. SCI, is currently Chair-Professor at the University of Karlsruhe in the School of Economics and Business Engineering and Professor Emeritus at the University of California. He is also the founder of Bravo Risk Management Group and Chief Scientist of FinAnalytica.

CHRISTIAN MENN, DR. RER. POL., is Hochschulassistent at the Chair of Statistics, Econometrics and Mathematical Finance at the University of Karlsruhe. Currently, he is a Visiting Scientist at the School of Operations Research and Industrial Engineering at Cornell University as a postdoctoral fellow.

FRANK J. FABOZZI, PhD, CFA, CPA, is the Frederick Frank Adjunct Professor of Finance at Yale University's School of Management. He is also a Fellow of the International Center for Finance at Yale University. Prior to joining the Yale faculty, Fabozzi was a visiting professor of finance in the Sloan School at MIT. Fabozzi has authored and edited many acclaimed books in finance and is also the Editor of the Journal of Portfolio Management.

Table of Contents

Preface	xi
About the Authors	xiii
CHAPTER 1 Introduction	1	(10)
Organization of the Book	5	(3)
References	8	(3)
PART ONE Probability and Statistics	11	(108)
CHAPTER 2 Discrete Probability Distributions	13	(10)
Basic Concepts	14	(1)
Discrete Probability Distributions Defined	14	(1)
Bernoulli Distribution	15	(1)
Binomial Distribution	15	(1)
Poisson Distribution	16	(5)
References	21	(2)
CHAPTER 3 Continuous Probability Distributions	23	(24)
Continuous Random Variables and Probability Distributions	23	(6)
The Normal Distribution	29	(3)
Other Popular Distributions	32	(14)
References	46	(1)
CHAPTER 4 Describing a Probability Distribution Function: Statistical Moments and Quantiles	47	(10)
Location	47	(1)
Dispersion	48	(1)
Asymmetry	48	(1)
Concentration in Tails	48	(1)
Statistical Moments	49	(1)
Quantiles	50	(3)
Sample Moments	53	(2)
Normal Distribution Revisited	55	(1)
References	56	(1)
CHAPTER 5 Joint Probability Distributions	57	(14)
Joint Probability Distributions Defined	57	(1)
Marginal Distributions	58	(1)
Dependence of Random Variables	59	(1)
Multivariate Normal Distribution	60	(7)
Elliptical Distributions	67	(3)
References	70	(1)
CHAPTER 6 Copulas	71	(10)
Drawbacks of Correlation	71	(1)
Overcoming the Drawbacks of Correlation: Copulas	72	(1)
Mathematical Definition of Copulas	73	(1)
References	74	(7)
CHAPTER 7 Stable Distributions	81	(12)
Properties of the Stable Distribution	82	(3)
Considerations in the Use of the Stable Distribution	85	(4)
Truncated Stable Distributions	89	(3)
References	92	(1)
CHAPTER 8 Estimation Methodologies	93	(26)
Fitting Probability Distributions by Maximum Likelihood Estimation	93	(3)
Confidence Bounds	96	(5)
Hypothesis Tests and P-Value	101	(4)
Relationship between Hypothesis Tests and Confidence Bounds	105	(4)
Fitting Stable Distributions	109	(2)
Comparing Probability Distributions: Testing for the Goodness of Fit	111	(6)
References	117	(2)
PART TWO Stochastic Processes	119	(42)
CHAPTER 9 Stochastic Processes in Discrete Time and Time Series Analysis	121	(22)
Stochastic Processes in Discrete Time	121	(9)
ARCH and GARCH Models	130	(3)
ARMA-GARCH Illustration	133	(7)
References	140	(3)
CHAPTER 10 Stochastic Processes in Continuous Time	143	(18)
The Poisson Process	144	(3)
Brownian Motion	147	(8)
Stochastic Differential Equations	155	(1)
Levy Processes	156	(2)
References	158	(3)
PART THREE Portfolio Selection	161	(52)
CHAPTER 11 Equity and Bond Return Distributions	163	(18)
Evidence from the U.S. Stock Market	163	(4)
Evidence from the U.S. Bond Market	167	(11)
References	178	(3)
CHAPTER 12 Risk Measures and Portfolio Selection	181	(18)
Desirable Features of Investment Risk Measures	181	(4)
Alternative Risk Measures for Portfolio Selection	185	(9)
References	194	(5)
CHAPTER 13 Risk Measures in Portfolio Optimization and Performance Measures	199	(14)
Efficient Frontiers and Return Distribution Assumption	200	(3)
Portfolio Optimization and Conditional Value-at-Risk versus Value-at-Risk	203	(3)
Performance Measures	206	(4)
References	210	(3)
PART FOUR Risk Management	213	(80)
CHAPTER 14 Market Risk	215	(38)
Adoption of VaR for Measuring Market Risk	216	(2)
VaR and Bank Capital Requirements	218	(4)
Computation of VaR	222	(6)
Evaluation of VaR Methods: Strengths and Weaknesses	228	(2)
Stable Modeling of VaR	230	(9)
Alternative to VaR: Expected Tail Loss	239	(7)
References	246	(3)
Appendix: Coherent Risk Measures	249	(4)
CHAPTER 15 Credit Risk	253	(30)
Credit Risk	253	(1)
Credit Risk Framework for Banks: Basel I and Basel II	254	(2)
Overview of Credit Risk Modeling	256	(7)
Credit Risk Management Tools	263	(4)
An Integrated Market and Credit Risk Management Framework Based on the Structural Approach	267	(5)
An Integrated Market and Credit Risk Management Framework Based on the Intensity-Based Model	272	(4)
Building An Econometric Model for the Intensity-Based Model	276	(2)
References	278	(5)
CHAPTER 16 Operational Risk	283	(10)
Operational Risk Defined	283	(3)
Capital Requirement for Operational Risk	286	(1)
Comparison of Market, Credit, and Operational Risk Distributions	287	(1)
Aggregated Stochastic Models for Operational Risk	288	(3)
References	291	(2)
PART FIVE Option Pricing	293	(60)
CHAPTER 17 Introduction to Option Pricing and the Binomial Model	295	(24)
Options Contracts	295	(2)
Basic Components of the Option Price	297	(1)
Boundary Conditions for the Price of an Option	298	(2)
Discrete-Time Option Pricing: Binomial Model	300	(12)
Convergence of the Binomial Model	312	(4)
References	316	(3)
CHAPTER 18 Black-Scholes Option Pricing Model	319	(18)
Motivation	319	(3)
Black-Scholes Formula	322	(1)
Computing a Call Option Price	323	(2)
Sensitivity of Option Price to a Change in Factors: The Greeks	325	(6)
Computing a Put Option Price	331	(1)
Assumptions Underlying the Black-Scholes Model and Basic Extensions	331	(3)
Black-Scholes Model Applied to the Pricing of Options on Bonds: Importance of Assumptions	334	(2)
References	336	(1)
CHAPTER 19 Extension of the Black-Scholes Model and Alternative Approaches	337	(16)
The "Smile Effect"	337	(2)
Continuous-Time Models	339	(6)
Discrete-Time Models	345	(4)
References	349	(4)
INDEX	353