Asset Price Dynamics, Volatility, and Prediction Book

Preface xiii

Chapter 1: Introduction 1
1.1 Asset Price Dynamics 1
1.2 Volatility 1
1.3 Prediction 2
1.4 Information 2
1.5 Contents 3
1.6 Software 5
1.7 Web Resources 6

PART I: Foundations 7

Chapter 2: Prices and Returns 9
2.1 Introduction 9
2.2 Two Examples of Price Series 9
2.3 Data-Collection Issues 10
2.4 Two Returns Series 13
2.5 Definitions of Returns 14
2.6 Further Examples of Time Series of Returns 19

Chapter 3: Stochastic Processes: Definitions and Examples 23
3.1 Introduction 23
3.2 Random Variables 24
3.3 Stationary Stochastic Processes 30
3.4 Uncorrelated Processes 33
3.5 ARMA Processes 36
3.6 Examples of ARMA 1 1 Specifications 44
3.7 ARIMA Processes 46
3.8 ARFIMA Processes 46
3.9 Linear Stochastic Processes 48
3.10 Continuous-Time Stochastic Processes 49
3.11 Notation for Random Variables and Observations 50

Chapter 4: Stylized Facts for Financial Returns 51
4.1 Introduction 51
4.2 Summary Statistics 52
4.3 Average Returns and Risk Premia 53
4.4 Standard Deviations 57
4.5 Calendar Effects 59
4.6 Skewness and Kurtosis 68
4.7 The Shape of the Returns Distribution 69
4.8 Probability Distributions for Returns 73
4.9 Autocorrelations of Returns 76
4.10 Autocorrelations of Transformed Returns 82
4.11 Nonlinearity of the Returns Process 92
4.12 Concluding Remarks 93
4.13 Appendix: Autocorrelation Caused by Day-of-the-Week Effects 94
4.14 Appendix: Autocorrelations of a Squared Linear Process 95
PART II: Conditional Expected Returns 97
Chapter 5: The Variance-Ratio Test of the Random Walk Hypothesis 99
5.1 Introduction 99
5.2 The Random Walk Hypothesis 100
5.3 Variance-Ratio Tests 102
5.4 An Example of Variance-Ratio Calculations 105
5.5 Selected Test Results 107
5.6 Sample Autocorrelation Theory 112
5.7 Random Walk Tests Using Rescaled Returns 115
5.8 Summary 120

Chapter 6: Further Tests of the Random Walk Hypothesis 121
6.1 Introduction 121
6.2 Test Methodology 122
6.3 Further Autocorrelation Tests 126
6.4 Spectral Tests 130
6.5 The Runs Test 133
6.6 Rescaled Range Tests 135
6.7 The BDS Test 136
6.8 Test Results for the Random Walk Hypothesis 138
6.9 The Size and Power of Random Walk Tests 144
6.10 Sources of Minor Dependence in Returns 148
6.11 Concluding Remarks 151
6.12 Appendix: the Correlation between Test Values for Two Correlated Series 153
6.13 Appendix: Autocorrelation Induced by Rescaling Returns 154

Chapter 7: Trading Rules and Market Efficiency 157
7.1 Introduction 157
7.2 Four Trading Rules 158
7.3 Measures of Return Predictability 163
7.4 Evidence about Equity Return Predictability 166
7.5 Evidence about the Predictability of Currency and Other Returns 168
7.6 An Example of Calculations for the Moving-Average Rule 172
7.7 Efficient Markets: Methodological Issues 175
7.8 Breakeven Costs for Trading Rules Applied to Equities 176
7.9 Trading Rule Performance for Futures Contracts 179
7.10 The Efficiency of Currency Markets 181
7.11 Theoretical Trading Profits for Autocorrelated Return Processes 184
7.12 Concluding Remarks 186

PART III: Volatility Processes 187

Chapter 8: An Introduction to Volatility 189
8.1 Definitions of Volatility 189
8.2 Explanations of Changes in Volatility 191
8.3 Volatility and Information Arrivals 193
8.4 Volatility and the Stylized Facts for Returns 195
8.5 Concluding Remarks 196

Chapter 9: ARCH Models: Definitions and Examples 197
9.1 Introduction 197
9.2 ARCH(1) 198
9.3 GARCH 1 1 199
9.4 An Exchange Rate Example of the GARCH 1 1 Model 205
9.5 A General ARCH Framework 212
9.6 Nonnormal Conditional Distributions 217
9.7 Asymmetric Volatility Models 220
9.8 Equity Examples of Asymmetric Volatility Models 222
9.9 Summary 233

Chapter 10: ARCH Models: Selection and Likelihood Methods 235
10.1 Introduction 235
10.2 Asymmetric Volatility: Further Specifications and Evidence 235
10.3 Long Memory ARCH Models 242
10.4 Likelihood Methods 245
10.5 Results from Hypothesis Tests 251
10.6 Model Building 256
10.7 Further Volatility Specifications 261
10.8 Concluding Remarks 264
10.9 Appendix: Formulae for the Score Vector 265
Chapter 11: Stochastic Volatility Models 267
11.1 Introduction 267
11.2 Motivation and Definitions 268
11.3 Moments of Independent SV Processes 270
11.4 Markov Chain Models for Volatility 271
11.5 The Standard Stochastic Volatility Model 278
11.6 Parameter Estimation for the Standard SV Model 283
11.7 An Example of SV Model Estimation for Exchange Rates 288
11.8 Independent SV Models with Heavy Tails 291
11.9 Asymmetric Stochastic Volatility Models 293
11.10 Long Memory SV Models 297
11.11 Multivariate Stochastic Volatility Models 298
11.12 ARCH versus SV 299
11.13 Concluding Remarks 301
11.14 Appendix: Filtering Equations 301

PART IV: High-Frequency Methods 303

Chapter 12: High-Frequency Data and Models 305
12.1 Introduction 305
12.2 High-Frequency Prices 306
12.3 One Day of High-Frequency Price Data 309
12.4 Stylized Facts for Intraday Returns 310
12.5 Intraday Volatility Patterns 316
12.6 Discrete-Time Intraday Volatility Models 321
12.7 Trading Rules and Intraday Prices 325
12.8 Realized Volatility: Theoretical Results 327
12.9 Realized Volatility: Empirical Results 332
12.10 Price Discovery 342
12.11 Durations 343
12.12 Extreme Price Changes 344
12.13 Daily High and Low Prices 346
12.14 Concluding Remarks 348
12.15 Appendix: Formulae for the Variance of the Realized Volatility Estimator 349

PART V: Inferences from Option Prices 351

Chapter 13: Continuous-Time Stochastic Processes 353
13.1 Introduction 353
13.2 The Wiener Process 354
13.3 Diffusion Processes 355
13.4 Bivariate Diffusion Processes 359
13.5 Jump Processes 361
13.6 Jump-Diffusion Processes 363
13.7 Appendix: a Construction of the Wiener Process 366

Chapter 14: Option Pricing Formulae 369
14.1 Introduction 369
14.2 Definitions, Notation, and Assumptions 370
14.3 Black-Scholes and Related Formulae 372
14.4 Implied Volatility 378
14.5 Option Prices when Volatility Is Stochastic 383
14.6 Closed-Form Stochastic Volatility Option Prices 388
14.7 Option Prices for ARCH Processes 391
14.8 Summary 394
14.9 Appendix: Heston's Option Pricing Formula 395

Chapter 15: Forecasting Volatility 397
15.1 Introduction 397
15.2 Forecasting Methodology 398
15.3 Two Measures of Forecast Accuracy 401
15.4 Historical Volatility Forecasts 403
15.5 Forecasts from Implied Volatilities 407
15.6 ARCH Forecasts that Incorporate Implied Volatilities 410
15.7 High-Frequency Forecasting Results 414
15.8 Concluding Remarks 420

Chapter 16: Density Prediction for Asset Prices 423
16.1 Introduction 423
16.2 Simulated Real-World Densities 424
16.3 Risk-Neutral Density Concepts and Definitions 428
16.4 Estimation of Implied Risk-Neutral Densities 431
16.5 Parametric Risk-Neutral Densities 435
16.6 Risk-Neutral Densities from Implied Volatility Functions 446
16.7 Nonparametric RND Methods 448
16.8 Towards Recommendations 450
16.9 From Risk-Neutral to Real-World Densities 451
16.10 An Excel Spreadsheet for Density Estimation 458
16.11 Risk Aversion and Rational RNDs 461
16.12 Tail Density Estimates 464
16.13 Concluding Remarks 465

Symbols 467
References 473
Author Index 503
Subject Index 513