INSEAD Library

Includes bibliographical references and index

Digitized

Modelling Financial Time Series Contents Preface to the 2nd edition Preface to the 1st edition 1 INTRODUCTION 1.1 Financial time series 1.2 About this study 1.3 The world's major financial markets 1.4 Examples of daily price series 1.5 A selective review of previous research Important questions The random walk hypothesis The efficient market hypothesis 1.6 Daily returns 1.7 Models 1.8 Models in this book 1.9 Stochastic processes General remarks Stationary processes Autocorrelation Spectral density White noise ARMA processes Gaussian processes 1.10 Linear stochastic processes Their definition Autocorrelation tests 2 FEATURES OF FINANCIAL RETURNS 2.1 Constructing financial time series Sources Time scales Additional information Using futures contracts Page xv xxv 1 1 2 3 4 8 8 8 10 12 13 15 16 16 16 17 18 19 20 23 23 23 24 26 26 26 27 27 28 2.2 Prices studied Spot prices Futures prices Commodity futures Financial futures Extended series 2.3 Average returns and risk premia Annual expected returns Common stocks and ordinary shares Spot commodities Spot currencies Commodity futures 2.4 Standard deviations Risks compared Futures and contract age 2.5 Calendar effects Day-of-the-week Stocks Currencies Agricultural futures Standard deviations Month-of-the-year effects for stocks 2.6 Skewness 2.7 Kurtosis 2.8 Plausible distributions 2.9 Autocorrelation First-lag Lags 1 to 30 Tests 2.10 Non-linear structure Not strict white noise A characteristic of returns Not linear Consequences of non-linear structure 2.11 Summary Appendix 2(A) Autocorrelation caused by day-of-the-week effects Appendix 2(B) Autocorrelations of a squared linear process 3 MODELLING PRICE VOLATILITY 3.1 Introduction 3.2 Elementary variance models Step change, discrete distributions Markov variances, discrete distributions 28 28 30 30 31 32 32 33 35 36 36 36 38 39 40 41 41 41 41 42 42 43 44 44 45 48 49 50 50 52 52 52 56 57 58 58 60 62 62 63 63 64 Step variances, continuous distributions Markov variances, continuous distributions 3.3 A general variance model Notation 3.4 Modelling variance jumps 3.5 Modelling frequent variance changes not caused by prices General models Stationary models The lognormal, autoregressive model 3.6 Modelling frequent variance changes caused by past prices General concepts Caused by past squared returns Caused by past absolute returns ARMACH models 3.7 Modelling autocorrelation and variance changes Variances not caused by returns Variances caused by returns 3.8 Parameter estimation for variance models 3.9 Parameter estimates for product processes Lognormal AR(1) Results 3.10 Parameter estimates for ARMACH processes Results 3.11 Summary Appendix 3(A) Results for ARCH processes 4 FORECASTING STANDARD DEVIATIONS 4.1 Introduction 4.2 Key theoretical results Uncorrelated returns Correlated returns Relative mean square errors Stationary processes 4.3 Forecasts: methodology and methods Benchmark forecast Parametric forecasts Product process forecasts ARMACH forecasts EWMA forecasts Futures forecasts Empirical RMSE 4.4 Forecasting results Absolute returns 65 66 67 69 69 70 70 72 73 75 75 76 78 78 79 81 82 83 84 86 88 90 92 93 95 97 97 98 98 100 100 100 101 101 101 102 103 103 104 105 106 106 Conditional standard deviations Two leading forecasts More distant forecasts Conclusions about stationarity Another approach 4.5 Recommended forecasts for the next day Examples 4.6 Summary 5 THE ACCURACY OF AUTOCORRELATION ESTIMATES 5.1 Introduction 5.2 Extreme examples 5.3 A special null hypothesis 5.4 Estimates of the variances of sample autocorrelations 5.5 Some asymptotic results Linear processes Non-linear processes 5.6 Interpreting the estimates 5.7 The estimates for returns 5.8 Accurate autocorrelation estimates Rescaled returns Variance estimates for recommended coefficients Exceptional series 5.9 Simulation results 5.10 Autocorrelations of rescaled processes 5.11 Summary 6 TESTING THE RANDOM WALK HYPOTHESIS 6.1 Introduction 6.2 Test methodology 6.3 Distributions of sample autocorrelations Asymptotic limits Finite samples 6.4 A selection of test statistics Autocorrelation tests Spectral tests The runs test 6.5 The price-trend hypothesis Price-trend autocorrelations An example Price-trend spectral density 6.6 Tests for random walks versus price-trends 6.7 Consequences of data errors 107 108 108 110 110 110 113 114 116 116 117 118 119 120 121 122 123 124 126 127 128 130 130 131 132 133 133 134 135 136 136 137 137 138 140 141 141 142 143 143 145 6.8 Results of random walk tests Stocks Commodities and currencies About the rest of this chapter 6.9 Some test results for returns 6.10 Power comparisons 6.11 Testing equilibrium models Stocks Simulation results Tests Other equilibrium models Conclusion 6.12 Institutional effects Limit rules Bid­ask spreads 6.13 Results for subdivided series 6.14 Conclusions 6.15 Summary Appendix 6(A) Correlation between test values for two related series 7 FORECASTING TRENDS IN PRICES 7.1 Introduction 7.2 Price-trend models A non-linear trend model A linear trend model 7.3 Estimating the trend parameters Methods Futures Spots Accuracy 7.4 Some results from simulations Estimates A puzzle solved 7.5 Forecasting returns: theoretical results The next return More distant returns Sums of future returns 7.6 Empirical forecasting results Benchmark forecasts Price-trend forecasts Summary statistics Futures Spots 146 150 152 156 157 159 161 161 163 165 166 166 167 167 169 169 170 172 172 174 174 174 176 176 178 178 179 181 183 183 183 185 185 186 187 187 188 188 189 189 190 192 7.7 Further forecasting theory Expected changes in prices Forecasting the direction of the trend Forecasting prices 7.8 Summary 8 EVIDENCE AGAINST THE EFFICIENCY OF FUTURES MARKETS 8.1 Introduction 8.2 The efficient market hypothesis 8.3 Problems raised by previous studies Filter rules Benchmarks Significance Optimization 8.4 Problems measuring risk and return Returns Risk Necessary assumptions 8.5 Trading conditions 8.6 Theoretical analysis Trading strategies Assumptions Conditions for trading profits Inefficient regions Some implications 8.7 Realistic strategies and assumptions Strategies Assumptions Notes on objectives 8.8 Trading simulated contracts Commodities Currencies 8.9 Trading results for futures Calibration contracts Test contracts Portfolio results 8.10 Towards conclusions 8.11 Summary 9 VALUING OPTIONS 9.1 Introduction 9.2 Black--Scholes option pricing formulae 9.3 Evaluating standard formulae 193 193 194 194 194 196 196 197 199 199 200 201 201 201 201 202 203 203 204 204 205 206 207 209 210 211 212 213 213 214 215 216 216 217 222 223 224 225 225 226 227 9.4 Call values when conditional variances change Formulae for a stationary process Examples Non-stationary processes Conclusions 9.5 Price trends and call values A formula for trend models Examples 9.6 Summary 10 CONCLUDING REMARKS 10.1 Price behaviour 10.2 Advice to traders 10.3 Further research 10.4 Stationary models Random walks Price trends APPENDIX: A COMPUTER PROGRAM FOR MODELLING FINANCIAL TIME SERIES Output produced Computer time required User-defined parameters Optional parameters Input requirements About the subroutines FORTRAN program References Author index Subject index 228 228 230 233 233 234 234 235 237 238 238 239 240 241 241 242 243 243 244 244 245 245 247 248 256 262 264