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  • Forecasting Volatility and Correlation: The Role of

    Option Implied Measures

    Christopher Andrew Coleman-Fenn

    B. Bus. Hons. (Banking and Finance)

    Grad. Dip. Sci. (Mathematics)

    Supervisor: Professor Adam Clements

    March 6, 2012

    Submitted as partial requirement

    for the degree of Doctor of Philosophy

    Queensland University of Technology

    School of Economics and Finance

    Brisbane, Australia

  • Keywords

    Volatility risk premium

    Implied volatility

    Implied correlation

    Model confidence set

    Intraday volatility

    Equicorrelation

    Realised equicorrelation

    i

  • Abstract

    Forecasts of volatility and correlation are important inputs into many practical

    financial problems. Broadly speaking, there are two ways of generating fore-

    casts of these variables. Firstly, time-series models apply a statistical weighting

    scheme to historical measurements of the variable of interest. The alternative

    methodology extracts forecasts from the market traded value of option con-

    tracts. An efficient options market should be able to produce superior forecasts

    as it utilises a larger information set of not only historical information but also

    the market equilibrium expectation of options market participants. While much

    research has been conducted into the relative merits of these approaches, this

    thesis extends the literature along several lines through three empirical studies.

    Firstly, it is demonstrated that there exist statistically significant benefits to

    taking the volatility risk premium into account for the implied volatility for

    the purposes of univariate volatility forecasting. Secondly, high-frequency op-

    tion implied measures are shown to lead to superior forecasts of the intraday

    stochastic component of intraday volatility and that these then lead on to supe-

    rior forecasts of intraday total volatility. Finally, the use of realised and option

    implied measures of equicorrelation are shown to dominate measures based on

    daily returns.

    ii

  • Contents

    1 Introduction 4

    1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

    1.2 Key Research Questions . . . . . . . . . . . . . . . . . . . . . . . 6

    1.3 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

    1.4 The Contributions of this Thesis . . . . . . . . . . . . . . . . . . 11

    2 Literature Review 13

    2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    2.2 Defining and Measuring Volatility . . . . . . . . . . . . . . . . . . 14

    2.2.1 Realised Volatility . . . . . . . . . . . . . . . . . . . . . . 16

    2.2.2 Stylised Facts of Volatility . . . . . . . . . . . . . . . . . . 28

    2.3 The Value of an Option . . . . . . . . . . . . . . . . . . . . . . . 35

    2.3.1 Black-Scholes-Merton Model . . . . . . . . . . . . . . . . 38

    2.3.2 Black-Scholes-Merton Implied Volatility . . . . . . . . . . 42

    2.3.3 Challenges to the Black-Scholes-Merton Model . . . . . . 44

    2.4 The Volatility Index . . . . . . . . . . . . . . . . . . . . . . . . . 50

    2.5 Forecast Performance of Implied Volatility . . . . . . . . . . . . . 55

    2.6 Univariate Time-series Forecasts . . . . . . . . . . . . . . . . . . 65

    iii

  • 2.6.1 GARCH class conditional volatility models . . . . . . . . 66

    2.6.2 Stochastic volatility models . . . . . . . . . . . . . . . . . 74

    2.6.3 Models for forecasting Realised Volatility . . . . . . . . . 78

    2.6.4 Hybrid Models . . . . . . . . . . . . . . . . . . . . . . . . 81

    2.7 Multivariate Time-Series Forecasts . . . . . . . . . . . . . . . . . 84

    2.7.1 Multivariate GARCH models . . . . . . . . . . . . . . . . 85

    2.7.2 Multivariate Stochastic Volatility Models . . . . . . . . . 92

    2.7.3 Multivariate Realised Volatility Models . . . . . . . . . . 93

    2.8 Implied Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 97

    2.9 Comparing Forecast Performance . . . . . . . . . . . . . . . . . . 100

    2.9.1 Regression based measures . . . . . . . . . . . . . . . . . 100

    2.9.2 Statistical loss functions . . . . . . . . . . . . . . . . . . . 101

    2.9.3 Distinguishing relative forecast performance . . . . . . . . 104

    3 Implied Volatility and the Volatility Risk Premium 111

    3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

    3.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

    3.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

    3.3.1 Model based forecasts . . . . . . . . . . . . . . . . . . . . 117

    3.3.2 A risk-adjusted VIX forecast . . . . . . . . . . . . . . . . 119

    3.3.3 Estimation of the Volatility Risk-Premium . . . . . . . . 121

    3.3.4 Evaluating forecasts . . . . . . . . . . . . . . . . . . . . . 123

    3.4 Empirical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

    3.4.1 22-day-ahead forecasts . . . . . . . . . . . . . . . . . . . . 127

    iv

  • 3.4.2 5-day-ahead forecasts . . . . . . . . . . . . . . . . . . . . 130

    3.4.3 1-day-ahead forecasts . . . . . . . . . . . . . . . . . . . . 132

    3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

    4 Forecasting Intraday Volatility: The Role of VIX Futures 136

    4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

    4.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

    4.2.1 An intraday volatility framework . . . . . . . . . . . . . . 146

    4.2.2 A semi-parametric framework . . . . . . . . . . . . . . . . 155

    4.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

    4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

    4.4.1 In-Sample Results . . . . . . . . . . . . . . . . . . . . . . 169

    4.4.2 Out-of-Sample Results . . . . . . . . . . . . . . . . . . . . 174

    4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

    5 Forecasting Equicorrelation 182

    5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

    5.2 General Framework and Models Considered . . . . . . . . . . . . 190

    5.2.1 The Linear Dynamic Equicorrelation Model . . . . . . . . 193

    5.2.2 Incorporating Implied Equicorrelation . . . . . . . . . . . 201

    5.3 Forecast Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 205

    5.3.1 Generating Forecasts . . . . . . . . . . . . . . . . . . . . . 206

    5.3.2 Statistical Evaluation of Forecasts . . . . . . . . . . . . . 208

    5.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

    5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

    v

  • 5.5.1 In-Sample Estimation Results . . . . . . . . . . . . . . . . 213

    5.5.2 Out-of-sample Forecast Results . . . . . . . . . . . . . . . 221

    5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

    6 Conclusion 227

    A Appendix 232

    A.1 Decomposition of Expectation of Squared Realised Volatility . . 232

    A.2 Statistical loss Functions and their derivatives . . . . . . . . . . 233

    A.3 Model Confidence Set results for remaining equicorrelation mea-

    sures, loss functions, and test statistics. . . . . . . . . . . . . . . 234

    vi

  • List of Tables

    3.1 MCS results for 22-day-ahead daily volatility forecasts . . . . . . 129

    3.2 MCS results for 5-day-ahead daily volatility forecasts . . . . . . . 131

    3.3 MCS results for 1-day-ahead daily volatility forecasts . . . . . . . 133

    4.1 In-sample estimation results for intraday volatility models . . . . 173

    4.2 MCS results for forecasts of q{t,i} . . . . . . . . . . . . . . . . . . 176

    4.3 MCS results for forecasts of r2{t,i} . . . . . . . . . . . . . . . . . . 179

    5.1 Xt measures descriptive statistics . . . . . . . . . . . . . . . . . . 212

    5.2 In-sample estimation results of equicorrelation models . . . . . . 215

    5.3 Vuong statistics for Xt measures . . . . . . . . . . . . . . . . . . 216

    5.4 Vuong statistics for restricted against unrestricted models . . . . 218

    5.5 MCS results for t forecasts; MSE, TR, Xt = REC . . . . . . . . 226

    A.1 Statistical loss functions and their derivatives . . . . . . . . . . . 233

    A.2 MCS results for t forecasts; MSE, TSq, Xt = REC . . . . . . . . 235

    A.3 MCS results for t forecasts; QLIKE, TR, Xt = REC . . . . . . . 236

    A.4 MCS results for t forecasts; QLIKE, TSq, Xt = REC . . . . . . . 237

    A.5 MCS results for t forecasts; MSE, TR, Xt = DREC . . . . . . . 238

    vii

  • A.6 MCS results for t forecasts; MSE, TSq, Xt = DREC . . . . . . . 239

    A.7 MCS results for t forecasts; QLIKE, TR, Xt = DREC . . . . . . 240

    A.8 MCS results for t forecasts; QLIKE, TSq, Xt = DREC . . . . . . 241

    A.9 MCS results for t forecasts; MSE, TR, Xt = SREC . . . . . . . . 242

    A.10 MCS results for t forecasts; MSE, TSq, Xt = SREC . . . . . . . 243

    A.11 MCS results for t forecasts; QLIKE, TR, Xt = SREC . . . . . . 244

    A.12 MCS results for t forecasts; QLIKE, TSq, Xt = SREC . . . . . . 245

    A.13 MCS results for t forecasts; MSE, TR, Xt = ut . . . . . . . . . . 246

    A.14 MCS results for t forecasts; MSE, TSq, Xt = ut . . . . . . . . . . 247

    A.15 MCS results for t forecasts; QLIKE, TR, Xt = ut . . . . . . . . . 248

    A.16 MCS results for t forecasts; QLIKE, TSq, Xt = ut . . . . . . . . 249

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