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1

Dedication

To Elsa, Reginald and Walter



Contents

Preface xv

Acknowledgements xvii

Part I: Volatility and Correlation Analysis

Chapter 1: Understanding Volatility and Correlation 3

1.1 The Statistical Nature of Volatility and Correlation 4

1.2 Volatility and Correlation in Financial Markets 9

1.3 Constant and Time-Varying Volatility Models 12

1.4 Constant and Time-Varying Correlation Models 14

1.5 Remarks on Implementing Volatility and Correlation Models 17

1.6 Summary 18

Chapter 2: Implied Volatility and Correlation 21

2.1 Understanding Implied Volatility* 22

2.1.1 Volatility in a Black-Scholes World 23

2.1.2 Call and Put Implied Volatilities 26

2.1.3 Differences between Implied and Statistical Volatilities 28

2.2 Features of Implied Volatility* 30

2.2.1 Smiles and Skews 30

2.2.2 Volatility Term Structures 31

2.2.3 Volatility Surfaces 32

2.3 The Relationship between Prices and Implied Volatility 34

2.3.1 Equity Prices and Volatility Regimes 34

2.3.2 Scenario Analysis of Prices and Implied Volatility 38

2.3.3 Implications for Delta Hedging 43

2.4 Implied Correlation 45

Chapter 3: Moving Average Models 49

3.1 Historic Volatility and Correlation* 50

3.1.1 Definition and Application 50

3.1.2 Historic Volatility in Financial Markets 52

3.1.3 Historic Correlation in Energy Markets 54

3.1.4 When and How Should Historic Estimates Be Used? 56

An asterisk * denotes that illustrative software is on the CD. The password for the CD is available from http:;.www.wiley.co.uk marketmodels.



3.2 Exponentially Weighted Moving Averages* 57

3.3 Constant Volatility and the Square Root of Time Rule 61

Chapter 4: GARCH Models 63

4.1 Introduction to Generalized Autoregressive Conditional Heteroscedasticity 65

4.1.1 Volatility Clustering 65

4.1.2 The Leverage Effect 68

4.1.3 The Conditional Mean and Conditional Variance Equations 69

4.2 A Survey of Univariate GARCH Models 70

4.2.1 ARCH 71

4.2.2 Symmetric GARCH* 72

4.2.3 Integrated GARCH and the Components Model 75

4.2.4 Asymmetric GARCH* 79

4.2.5 GARCH Models for High-Frequency Data 82

4.3 Specification and Estimation of GARCH Models 84

4.3.1 Choice of Data, Stability of GARCH Parameters

and Long-Term Volatility 84

4.3.2 Parameter Estimation Algorithms 94

4.3.3 Estimation Problems 96

4.3.4 Choosing the Best GARCH Model 96

4.4 Applications of GARCH Models 97

4.4.1 GARCH Volatility Term Structures* 98

4.4.2 Option Pricing and Hedging 103

4.4.3 Smile Fitting 106

4.5 Multivariate GARCH 107

4.5.1 Time-Varying Correlation 108

4.5.2 Multivariate GARCH Parameterizations 112

4.5.3 Time-Varying Co variance Matrices Based on

Univariate GARCH Models 114

Chapter 5: Forecasting Volatility and Correlation 117

5.1 Evaluating the Accuracy of Point Forecasts 119

5.1.1 Statistical Criteria 121

5.1.2 Operational Criteria 124

5.2 Confidence Intervals for Volatility Forecasts 126

5.2.1 Moving Average Models 126

5.2.2 GARCH Models 128

5.2.3 Confidence Intervals for Combined Forecasts 128

5.3 Consequences of Uncertainty in Volatility and Correlation 135

5.3.1 Adjustment in Mark-to-Model Value of an Option* 135

5.3.2 Uncertainty in Dynamically Hedged Portfolios 138

Part II: Modelling the Market Risk of Portfolios

Chapter 6: Principal Component Analysis 143

6.1 Mathematical Background 145



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