Fat-Tailed and Skewed Asset Return

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.

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.

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

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

Summary

Author Biography

Table of Contents

Supplemental Materials