Part Ⅰ DATA EXPLORATION,ESTIMATION AND SIMULATION 3
1 UNIVARIATE EXPLORATORY DATA ANALYSIS 3
1.1 Data,Random Variables and Their Distributions 3
1.1.1 The PCS Data 4
1.1.2 The S＆P 500 Index and Financial Returns 5
1.1.3 Random Variables and Their Distributions 7
1.1.4 Examples of Probability Distribution Families 8
1.2 First Exploratory Data Analysis Tools 13
1.2.1 Random Samples 13
1.2.2 Histograms 14
1.3 More Nonparametric Density Estimation 16
1.3.1 Kernel Density Estimation 17
1.3.2 Comparison with the Histogram 19
1.3.3 S＆P Daily Returns 19
1.3.4 Importance of the Choice of the Bandwidth 22
1.4 Quantiles and Q-Q Plots 23
1.4.1 Understanding the Meaning of Q-Q Plots 24
1.4.2 Value at Risk and Expected Shortfall 25
1.5 Estimation from Empirical Data 28
1.5.1 The Empirical Distribution Function 28
1.5.2 Order Statistics 29
1.5.3 Empirical Q-Q Plots 30
1.6 Random Generators and Monte Carlo Samples 31
1.7 Extremes and Heavy Tail Distributions 35
1.7.1 S＆P Daily Returns,Once More 35
1.7.2 The Example of the PCS Index 37
1.7.3 The Example of the Weekly S＆P Returns 41
Problems 43
Notes ＆ Complements 46
2 MULTIVARIATE DATA EXPLORATION 49
2.1 Multivariate Data and First Measure of Dependence 49
2.1.1 Density Estimation 51
2.1.2 The Correlation Coefficient 53
2.2 The Multivariate Normal Distribution 56
2.2.1 Simulation of Random Samples 57
2.2.2 The Bivariate Case 58
2.2.3 A Simulation Example 59
2.2.4 Let's Have Some Coffee 60
2.2.5 Is the Joint Distribution Normal? 62
2.3 Marginals and More Measures of Dependence 63
2.3.1 Estimation of the Coffee Log-Return Distributions 64
2.3.2 More Measures of Dependence 68
2.4 Copulas and Random Simulations 70
2.4.1 Copulas 71
2.4.2 First Examples of Copula Families 72
2.4.3 Copulas and General Bivariate Distributions 74
2.4.4 Fitting Copulas 76
2.4.5 Monte Carlo Simulations with Copulas 77
2.4.6 A Risk Management Example 80
2.5 Principal Component Analysis 84
2.5.1 Identification of the Principal Components of a Data Set 84
2.5.2 PCA with S-Plus 87
2.5.3 Effective Dimension of the Space of Yield Curves 87
2.5.4 Swap Rate Curves 90
Appendix 1:Calculus with Random Vectors and Matrices 92
Appendix 2:Families ofCopulas 95
Problems 98
Notes ＆ Complements 101
Part Ⅱ REGRESSION 105
3 PARAMETRIC REGRESSION 105
3.1 Simple Linear Regression 105
3.1.1 Getting the Data 106
3.1.2 First Plots 107
3.1.3 Regression Set-up 108
3.1.4 Simple Linear Regression 111
3.1.5 Cost Minimizations 114
3.1.6 Regression as a Minimization Problem 114
3.2 Regression for Prediction ＆ Sensitivities 116
3.2.1 Prediction 116
3.2.2 Introductory Discussion of Sensitivity and Robustness 118
3.2.3 Comparing L2 and L1 Regressions 119
3.2.4 Taking Another Look at the Coffee Data 121
3.3 Smoothing versus Distribution Theory 123
3.3.1 Regression and Conditional Expectation 123
3.3.2 Maximum Likelihood Approach 124
3.4 Multiple Regression 129
3.4.1 Notation 129
3.4.2 The S-Plus Function lm 130
3.4.3 R2 as a Regression Diagnostic 131
3.5 Matrix Formulation and Linear Models 133
3.5.1 Linear Models 134
3.5.2 Least Squares(Linear)Regression Revisited 134
3.5.3 First Extensions 139
3.5.4 Testing the CAPM 142
3.6 Polynomial Regression 145
3.6.1 Polynomial Regression as a Linear Model 146
3.6.2 Example of S-Plus Commands 146
3.6.3 Important Remark 148
3.6.4 Prediction with Polynomial Regression 148
3.6.5 Choice of the Degree p 150
3.7 Nonlinear Regression 150
3.8 Term Structure of Interest Rates:A Crash Course 154
3.9 Parametric Yield Curve Estimation 160
3.9.1 Estimation Procedures 160
3.9.2 Practical Implementation 161
3.9.3 S-Plus Experiments 163
3.9.4 Concluding Remarks 165
Appendix:Cautionary Notes on Some S-Plus Idiosyncracies 166
Problems 169
Notes ＆ Complements 172
4 LOCAL ＆ NONPARAMETRIC REGRESSION 175
4.1 Review of the Regression Setup 175
4.2 Natural Splines as Local Smoothers 177
4.3 Nonparametric Scatterplot Smoothers 178
4.3.1 Smoothing Splines 179
4.3.2 Locally Weighted Regression 181
4.3.3 A Robust Smoother 182
4.3.4 The Super Smoother 183
4.3.5 The Kernel Smoother 183
4.4 More Yield Curve Estimation 186
4.4.1 A First Estimation Method 186
4.4.2 A Direct Application of Smoothing Splines 188
4.4.3 US and Japanese Instantaneous Forward Rates 188
4.5 Multivariate Kernel Regression 189
4.5.1 Running the Kernel in S-Plus 192
4.5.2 An Example Involving the June 1998 S＆P Futures Contract 193
4.6 Projection Pursuit Regression 197
4.6.1 The S-Plus Function ppreg 198
4.6.2 ppreg Prediction of the S&P Indicators 200
4.7 Nonparametric Option Pricing 205
4.7.1 Generalities on Option Pricing 205
4.7.2 Nonparametric Pricing Alternatives 212
4.7.3 Description of the Data 213
4.7.4 The Actual Experiment 214
4.7.5 Numerical Results 220
Appendix:Kernel Density Estimation ＆ Kernel Regression 222
Problems 225
Notes ＆ Complements 233
Part Ⅲ TIME SERIES ＆ STATE SPACE MODELS 239
5 TIME SERIES MODELS:AR,MA,ARMA,＆ ALL THAT 239
5.1 Notation and First Definitions 239
5.1.1 Notation 239
5.1.2 Regular Time Series and Signals 240
5.1.3 Calendar and Irregular Time Series 241
5.1.4 Example of Daily S＆P 500 Futures Contracts 243
5.2 High Frequency Data 245
5.2.1 TimeDate Manipulations 248
5.3 Time Dependent Statistics and Stationarity 253
5.3.1 Statistical Moments 253
5.3.2 The Notion of Stationarity 254
5.3.3 The Search for Stationarity 258
5.3.4 The Example of the CO2 Concentrations 261
5.4 First Examples of Models 263
5.4.1 White Noise 264
5.4.2 Random Walk 267
5.4.3 Auto Regressive Time Series 268
5.4.4 Moving Average Time Series 272
5.4.5 Using the Backward Shift Operator B 275
5.4.6 Linear Processes 276
5.4.7 Causality,Stationarity and Invertibility 277
5.4.8 ARMA Time Series 281
5.4.9 ARIMA Models 282
5.5 Fitting Models to Data 282
5.5.1 Practical Steps 282
5.5.2 S-Plus Implementation 284
5.6 Putting a Price on Temperature 289
5.6.1 Generalities on Degree Days 290
5.6.2 Temperature Options 291
5.6.3 Statistical Analysis of Temperature Historical Data 294
Appendix:More S-Plus Idiosyncracies 301
Problems 304
Notes ＆ Complements 308
6 MULTIVARIATE TIME SERIES,LINEAR SYSTEMS & KALMAN FILTERING 311
6.1 Multivariate Time Series 311
6.1.1 Stationarity and Auto-Covariance Functions 312
6.1.2 Multivariate White Noise 312
6.1.3 Multivariate AR Models 313
6.1.4 Backto Temperature Options 316
6.1.5 Multivariate MA ＆ ARIMA Models 318
6.1.6 Cointegration 319
6.2 State Space Models 321
6.3 Factor Models as Hidden Markov Processes 323
6.4 Kalman Filtering of Linear Systems 326
6.4.1 One-Step-Ahead Prediction 326
6.4.2 Derivation of the Recursive Filtering Equations 327
6.4.3 Writing an S Function for Kalman Prediction 329
6.4.4 Filtering 331
6.4.5 More Predictions 332
6.4.6 Estimation of the Parameters 333
6.5 Applications to Linear Models 335
6.5.1 State Space Representation of Linear Models 335
6.5.2 Linear Models with Time Varying Coefficients 336
6.5.3 CAPM with Time Varying β's 337
6.6 State Space Representation of Time Series 338
6.6.1 The Case of AR Series 339
6.6.2 The General Case of ARMA Series 341
6.6.3 Fitting ARMA Models by Maximum Likelihood 342
6.7 Example:Prediction of Quarterly Earnings 343
Problems 346
Notes ＆ Complements 351
7 NONLINEAR TIME SERIES:MODELS AND SIMULATION 353
7.1 First Nonlinear Time Series Models 353
7.1.1 Fractional Time Series 354
7.1.2 Nonlinear Auto-Regressive Series 355
7.1.3 Statistical Estimation 356
7.2 More Nonlinear Models:ARCH,GARCH ＆ All That 358
7.2.1 Motivation 358
7.2.2 ARCH Models 359
7.2.3 GARCH Models 361
7.2.4 S-Plus Commands 362
7.2.5 Fitting a GARCH Model to Real Data 363
7.2.6 Generalizations 371
7.3 Stochastic Volatility Models 373
7.4 Discretization of Stochastic Differential Equations 378
7.4.1 Discretization Schemes 379
7.4.2 Monte Carlo Simulations:A First Example 381
7.5 Random Simulation and Scenario Generation 383
7.5.1 A Simple Model for the S＆P 500 Index 383
7.5.2 Modeling the Short Interest Rate 386
7.5.3 Modeling the Spread 388
7.5.4 Putting Everything Together 389
7.6 Filtering of Nonlinear Systems 391
7.6.1 Hidden Markov Models 391
7.6.2 General Filtering Approach 392
7.6.3 Particle Filter Approximations 393
7.6.4 Filtering in Finance?Statistical Issues 396
7.6.5 Application:Tracking Volatility 397
Appendix:Preparing Index Data 403
Problems 404
Notes ＆ Complements 408
APPENDIX:AN INTRODUCTION TO S AND S-Plus 411
References 429
Notation Index 433
Data Set Index 435
S-Plus Index 437
Author Index 441
Subject Index 445