Forecasting Volatility in the Financial Markets (eBook)

Front Cover 1
Forecasting Volatility in the Financial Markets 4
Copyright Page 5
Table of Contents 6
List of contributors 8
Preface to Third Edition 10
Introduction 12
Chapter 1 Volatility modelling and forecasting in finance 14
1.1 Introduction 14
1.2 Autoregressive moving average models 15
1.3 Changes in volatility 16
1.3.1 Volatility in financial time series: stylized facts 16
1.3.2 The basic set-up 17
1.4 ARCH models 17
1.4.1 Generalized ARCH 18
1.4.2 Integrated ARCH 20
1.4.3 Exponential ARCH 21
1.4.4 ARCH-M model 22
1.4.5 Fractionally integrated ARCH 23
1.4.6 Other univariate ARCH formulations 23
1.4.7 Multivariate ARCH models 24
1.5 Stochastic variance models 24
1.5.1 From continuous time financial models to discrete time SV models 26
1.5.2 Persistence and the SV model 27
1.5.3 Long memory SV models 27
1.5.4 Risk-return trade-off in SV models 28
1.5.5 Multivariate SV models 28
1.6 Structural changes in the underlying process 29
1.6.1 Regime switching models 29
1.6.2 Extensions of the regime switching models 31
1.7 Threshold models 33
1.7.1 Self-exciting threshold models 33
1.7.2 Open loop threshold models 35
1.7.3 Closed loop threshold models 35
1.7.4 Smooth threshold autoregressive models 35
1.7.5 Identification in SETAR models 35
1.7.6 A threshold AR(1) model 36
1.7.7 A threshold MA model 37
1.7.8 Threshold models and asymmetries in volatility 38
1.7.9 Testing for non-linearity 40
1.7.10 Threshold estimation and prediction of TAR models 40
1.8 Volatility forecasting 41
1.8.1 Volatility forecasting based on time-series models 41
1.8.2 Volatility forecasting based on option ISD (Implied Standard Deviation) 44
1.9 Conclusion 45
References and further reading 46
Notes 57
Chapter 2 What good is a volatility model? 60
Abstract 60
2.1 Introduction 60
2.1.1 Notation 61
2.1.2 Types of volatility models 62
2.2 Stylized facts about asset price volatility 63
2.2.1 Volatility exhibits persistence 63
2.2.2 Volatility is mean reverting 64
2.2.3 Innovations may have an asymmetric impact on volatility 65
2.2.4 Exogenous variables may influence volatility 65
2.2.5 Tail probabilities 66
2.2.6 Forecast evaluation 66
2.3 An empirical example 67
2.3.1 Summary of the data 67
2.3.2 A volatility model 68
2.3.3 Mean reversion and persistence in volatility 69
2.3.4 An asymmetric volatility model 72
2.3.5 A model with exogenous volatility regressors 73
2.3.6 Aggregation of volatility models 73
2.4 Conclusions and challenges for future research 74
References 75
Notes 76
Chapter 3 Applications of portfolio variety 78
Abstract 78
3.1 Introduction 78
3.2 Some applications of variety 79
3.3 Empirical research on variety 80
3.4 Variety and risk estimation 82
3.5 Variety as an explanation of active management styles 83
3.6 Summary 84
References 85
Chapter 4 A comparison of the properties of realized variance for the FTSE 100 and FTSE 250 equity indices 86
4.1 Introduction 86
4.2 Data 88
4.3 Theory and empirical methodology 89
4.3.1 Realized variance 89
4.3.2 Optimal sampling frequency 91
4.3.3 Estimation 93
4.3.4 Forecasting 95
4.4 Initial data analysis 96
4.5 Estimation and forecasting results 99
4.5.1 GARCH/EGARCH estimation 99
4.5.2 GARCH/EGARCH forecasting 101
4.6 Conclusion 104
References 105
Notes 106
Appendices 107
Appendix 4.1: VBA code for RV computation 107
Appendix 4.2: EViews output for specification of mean equation 110
Chapter 5 An investigation of the relative performance of GARCH models versus simple rules in forecasting volatility 114
Abstract 114
5.1 Introduction 114
5.1.1 Measuring volatility 115
5.1.2 Features of volatility 116
5.2 Volatility forecasting methods 116
5.2.1 ARIMA models 116
5.2.2 ARCH/GARCH 119
5.3 Assessing forecasting performance 121
5.4 The performance of different forecasting models – empirical results 122
5.5 MSE of linear filter forecasts – theory 123
5.5.1 MSE of moving average methods 125
5.5.2 MSE under GARCH returns 125
5.6 Linear filter vs GARCH forecasts 126
5.6.1 MSE of linear filter forecasts 126
5.6.2 MSE of GARCH forecasts – simulation 127
5.6.3 Discussion of MSE results 128
5.6.4 The effects of leptokurtosis 130
5.7 Other methods of measuring true volatility 132
5.8 Empirical section – UK stock-market volatility 134
5.8.1 Data 134
5.8.2 Methodology 135
5.8.3 Results 136
5.8.4 Summary of results 139
5.9 Conclusion 140
References 140
Notes 142
Chapter 6 Stochastic volatility and option pricing 144
Summary 144
6.1 Introduction 144
6.2 The stochastic volatility (SV) model 145
6.2.1 The discrete-time stochastic autoregressive volatility model 145
6.2.2 The continuous-time stochastic volatility model 147
6.2.3 The jump-diffusion model with stochastic volatility 148
6.3 Option pricing under stochastic volatility 150
6.3.1 Pricing options under SV and jump: closed form solutions 151
6.3.2 Pricing options under SV and jump: Monte Carlo simulation 156
6.3.3 Implications of stochastic volatility and jumps on option prices 158
6.4 Estimation of stochastic volatility models 162
6.4.1 Indirect inference: a general estimation approach 163
6.4.2 Efficient method-of-moments (EMM): an efficient estimation approach 166
6.5 Forecasting stochastic volatility 169
6.5.1 Underlying volatility and implied volatility 169
6.5.2 Forecasting volatility based on standard volatility models 170
6.5.3 Forecasting underlying volatility using implied volatility 173
6.6 Conclusion 177
6.7 Appendix: derivation of option pricing formula for general jump-diffusion process with stochastic volatility and stochastic interest rates 177
References 179
Notes 184
Chapter 7 Modelling slippage: an application to the bund futures contract 186
7.1 Introduction 186
7.2 Data description 187
7.3 Slippage 191
7.4 Empirical evidence and econometric challenges 192
7.5 Conclusion 196
7.6 Appendix Regression results 197
References 198
Notes 199
Chapter 8 Real trading volume and price action in the foreign exchange markets 200
Summary 200
8.1 The currency market 201
8.2 Description of the EBS system 203
8.3 Empirical study of recorded transactions 204
8.4 Number of transactions, volume and volatility 205
8.5 Liquidity cost (Profit) 208
8.6 Liquidity cost and volume 209
8.7 Liquidity cost and time of the day 210
8.8 Final remarks 211
References 211
Notes 212
Chapter 9 Implied risk-neutral probability density functions from option prices: a central bank perspective 214
9.1 Introduction 214
9.2 The relationship between option prices and RND functions 216
9.3 The Black–Scholes formula and its RND function 216
9.4 The implied volatility smile curve 217
9.5 Estimating implied terminal RND functions 218
9.6 Application of the two-lognormal mixture approach to options on short-term interest rates 221
9.7 Monetary policy uses of the information contained in implied RND functions 222
9.7.1 Validating the two-lognormal mixture distribution approach 222
9.7.2 Assessing monetary conditions 227
9.7.3 Assessing monetary credibility 228
9.7.4 Assessing the timing and effectiveness of monetary operations 229
9.7.5 Identifying market anomalies 229
9.8 Using the implied RND function as a volatility forecasting tool 230
9.8.1 Calculating RND-based uncertainty measures 230
9.8.2 Results 233
9.9 Conclusions 234
References 235
Notes 237
Chapter 10 Hashing GARCH: a reassessment of volatility forecasting performance 240
Summary 240
10.1 Introduction 240
10.2 ARCH forecasts and properties 241
10.3 Forecasting performance evaluation 242
10.3.1 Statistics-based evaluation 243
10.3.2 Utility-based evaluation 245
10.3.3 Profit-based/preference-free evaluation 246
10.4 The pathology of ARCH forecast evaluation 247
10.4.1 A simulation experiment 249
10.5 Some general results 251
10.5.1 The compound normal 251
10.5.2 The Gram–Charlier class 252
10.6 Conclusions 254
10.7 Appendix 255
10.7.1 Mean and variance of ln(v2) 255
10.7.2 Proof of proposition 2 256
References 257
Notes 259
Chapter 11 Implied volatility forecasting: a comparison of different procedures including fractionally integrated models with applications to UK equity 262
Summary 262
11.1 Introduction 262
11.2 Data 263
11.3 Models for volatility 264
11.3.1 GARCH models 264
11.3.2 Log-ARFIMA models 265
11.3.3 Moving average methods 274
11.4 Out-of-sample forecasting performance tests 275
11.4.1 Forecasting procedure 275
11.4.2 Results 276
11.5 Conclusion 288
References 288
Notes 289
Chapter 12 GARCH predictions and the predictions of option prices 292
12.1 Prediction of GARCH models 292
12.2 Numerical results 296
12.3 Application to option pricing models 303
References 307
Notes 307
Chapter 13 Volatility forecasting in a tick data model 308
Summary 308
13.1 Introduction 308
13.2 The modelling framework 309
13.3 The functional form of v 310
13.4 Discussion and conclusions 312
References 312
Chapter 14 An econometic model of downside risk 314
Summary 314
14.1 Introduction 314
14.2 Overview of semivariance 315
14.2.1 Introduction to semivariance 315
14.2.2 Limitations of variance as a risk measure 316
14.2.3 Limitations of semivariance 317
14.2.4 Asymmetry of returns and semivariance 318
14.3 Risk modelling techniques 318
14.3.1 The ARCH class of models 319
14.4 Dynamic models of semivariance 322
14.4.1 ARCH-semivariance (ARCH-SV) 322
14.4.2 Extension of model and other issues 324
14.4.3 A generalized ARCH-SV model 325
14.4.4 Estimation of the model 326
14.4.5 Extensions of the GARCH-SV model (GARCH-SV(M)) 328
14.4.6 The differential impact of lagged volatility 328
14.4.7 Semivariance and asymmetric ARCH models 329
14.5 Application of the dynamic semivariance models 333
14.5.1 Data 333
14.5.2 Dynamic models of semivariance 334
14.5.3 Model extensions 338
14.6 Conclusion 342
14.7 Statistical appendix 342
References 343
Notes 344
Chapter 15 Variations in the mean and volatility of stock returns around turning points of the business cycle 346
Summary 346
15.1 Introduction 346
15.2 Variations in stock returns around turning points of the business cycle 347
15.2.1 Business cycle variation in expected excess returns 348
15.2.2 Assessing the effect of using ex-post information 351
15.2.3 Business cycle variation in stock market volatility 352
15.2.4 Monte Carlo experiments 355
15.3 Business cycle variation in the conditional distribution of stock returns 358
15.3.1 Negative expected excess returns around peaks of the business cycle 359
15.4 Conclusion 360
References 362
Notes 362
Chapter 16 Long memory in stochastic volatility 364
Summary 364
16.1 Introduction 364
16.2 Dynamic properties 366
16.3 Estimation and testing 369
16.4 Signal extraction and prediction 370
16.5 Extensions 374
16.6 Conclusions 374
16.7 Appendix 375
References 375
Notes 376
Chapter 17 GARCH processes – some exact results, some difficulties and a suggested remedy 378
Summary 378
17.1 Introduction 378
17.2 GARCH(1,1) and multiplicative GARCH(1,1) models 379
17.2.1 The GARCH(1,1) 379
17.2.2 MULGARCH(1,1) 382
17.3 An alternative GARCH(1,1) 386
17.4 GARCH(1,1) via a heterogeneous Poisson process 389
17.5 An empirical comparison 394
17.6 Appendix 395
References 401
Notes 402
Chapter 18 Generating composite volatility forecasts with random factor betas 404
Summary 404
18.1 Introduction 404
18.2 Random factor beta models 405
18.3 Composite volatility forecasts 408
18.3.1 Individual beta and volatility forecasts 410
18.3.2 Composite ARCBeta–GARCH forecasts 412
18.3.3 Latent AR–GARCH forecasts 413
18.4 Discussion and conclusions 416
18.5 Appendix 417
18.5.1 Proof of Lemma 1 417
References 418
Notes 419
Index 420