Tze Leung Lai/ Haipeng Xing Statistical Models and Methods for Financial Markets B 374756 4Q Springer
Preface \ vii Part I Basic Statistical Methods and Financial Applications 1 Linear Regression Models 3 1.1 Ordinary least squares (OLS) 4 1.1.1 Residuals and their sum of squares 4 1.1.2 Properties of projection matrices 5 1.1.3 Properties of nonnegative definite matrices 6 1.1.4 Statistical properties of OLS estimates 7 1.2 Statistical inference 8 1.2.1 Confidence intervals 8 1.2.2 ANOVA (analysis of variance) tests 10 1.3 Variable selection 12 1.3.1 Test-based and other variable selection criteria 12 1.3.2 Stepwise variable selection 15 1.4 Regression diagnostics 16 1.4.1 Analysis of residuals 17 1.4.2 Influence diagnostics 18 1.5 Extension to stochastic regressors 19 1.5.1 Minimum-variance linear predictors 19 1.5.2 Futures markets and hedging with futures contracts... 20 1.5.3 Inference in the case of stochastic regressors 21 1.6 Bootstrapping in regression " 22 1.6.1 The plug-in principle and bootstrap resampling 22 1.6.2 Bootstrapping regression models 24 1.6.3 Bootstrap confidence intervals 25 1.7 Generalized least squares 25
1.8 Implementation and illustration 26 Exercises 32 Multivariate Analysis and Likelihood Inference 37 2.1 Joint distribution of random variables 38 2.1.1 Change of variables 39 2.1.2 Mean and covariance matrix 39 2.2 Principal component analysis (PCA) 41 2.2.1 Basic definitions 41 2.2.2 Properties of principal components 42 2.2.3 An example: PCA of U.S. Treasury-LIBOR swap rates 44 2.3 Multivariate normal distribution 48 2.3.1 Definition and density function 48 2.3.2 Marginal and conditional distributions 50 2.3.3 Orthogonality and independence, with applications to regression 50 2.3.4 Sample covariance matrix and Wishart distribution... 52, 2.4 Likelihood inference 55 2.4.1 Method of maximum likelihood 55 2.4.2 Asymptotic inference 58 2.4.3 Parametric bootstrap 59 Exercises 60 Basic Investment Models and Their Statistical Analysis... 63 3.1 Asset returns 64 3.1.1 Definitions 64 3.1.2 Statistical models for asset prices and returns 66 3.2 Markowitz's portfolio theory 67 3.2.1 Portfolio weights 67 3.2.2 Geometry of efficient sets 68 3.2.3 Computation of efficient portfolios 69 3.2.4 Estimation of ft and S and an example 71 3.3 Capital asset pricing model (CAPM) 72 3.3.1 The model 72 3.3.2 Investment implications ; 77 3.3.3 Estimation and testing 77 3.3.4 Empirical studies of CAPM 79 3.4 Multifactor pricing models 81 3.4.1 Arbitrage pricing theory 81 3.4.2 Factor analysis 82 3.4.3 The PCA approach 85 3.4.4 The Fama-French three-factor model 86
xv 3.5 Applications of resampling to portfolio management 87 3.5.1 Michaud's resampled efficient frontier 87 3.5.2 Bootstrap estimates of performance 88 Exercises 89 Parametric Models and Bayesian Methods 93 4.1 Maximum likelihood and generalized linear models 94 4.1.1 Numerical methods for computing MLE 94 4.1.2 Generalized linear models 95 4.2 Nonlinear regression models 97 4.2.1 The Gauss-Newton algorithm 98 4.2.2 Statistical inference 100 4.2.3 Implementation and an example 101 4.3 Bayesian inference 103 4.3.1 Prior and posterior distributions 103 4.3.2 Bayes procedures 104 4.3.3 Bayes estimators of multivariate normal mean and covariance matrix 105 4.3.4 Bayes estimators in Gaussian regression models 107 4.3.5 Empirical Bayes and shrinkage estimators 108 4.4 Investment applications of shrinkage estimators and Bayesian methods 109 4.4.1 Shrinkage estimators of fi and for the plug-in efficient frontier 110 4.4.2 An alternative Bayesian approach Ill Exercises 113 Time Series Modeling and Forecasting 115 5.1 Stationary time series analysis 115 5.1.1 Weak stationarity 115 5.1.2 Tests of independence 117 5.1.3 Wold decomposition and MA, AR, and ARMA models. 119 5.1.4 Forecasting in ARMA models 121 5.1.5 Parameter estimation and order determination 122 5.2 Analysis of nonstationary time series 123 5.2.1 Detrending ' 123 5.2.2 An empirical example 124 5.2.3 Transformation and differencing 128 5.2.4 Unit-root nonstationarity and ARIMA models 129 5.3 Linear state-space models and Kalman filtering 130 5.3.1 Recursive formulas for P t t _i,x t t _i, and x t t 131 5.3.2 Dynamic linear models and time-varying betas in CAPM 133
xvi Contents Exercises 135 6 Dynamic Models of Asser Returns and Their Volatilities.. 139 6.1 Stylized facts on time series of asset returns 140 6.2 Moving average estimators of time-varying volatilities 144 6.3 Conditional heteroskedastic models 146 6.3.1 The ARCH model 146 6.3.2 The GARCH model 7 147 6.3.3 The integrated GARCH model 152 6.3.4 The exponential GARCH model 152 6.4 The ARMA-GARCH and ARMA-EGARCH models 155 6.4.1 Forecasting future returns and volatilities 156 6.4.2 Implementation and illustration 156 Exercises 157 Part II Advanced Topics in Quantitative Finance 7 Nonparametric Regression and Substantive-Empirical Modeling 163 7.1 Regression functions and minimum-variance prediction 164 7.2 Univariate predictors 165 7.2.1 Running-mean/running-line smoothers and local polynomial regression 165 7.2.2 Kernel smoothers 166 7.2.3 Regression splines 166 7.2.4 Smoothing cubic splines 169 7.3 Selection of smoothing parameter 170 7.3.1 The bias-variance trade-off 170 7.3.2 Cross-validation 171 7.4 Multivariate predictors 172 7.4.1 Tensor product basis and multivariate adaptive regression splines 172 7.4.2 Additive regression models 173 7.4.3 Projection pursuit regression 174 7.4.4 Neural networks 174 7.5 A modeling approach that combines domain knowledge with nonparametric regression 176 7.5.1 Penalized spline models and estimation of forward rates 177 7.5.2 A semiparametric penalized spline model for the forward rate curve of corporate debt 178 Exercises 179
xvii Option Pricing and Market Data 181 8.1 Option prices and pricing theory 182 8.1.1 Options data and put-call parity 182 8.1.2 The Black-Scholes formulas for European options 183 8.1.3 Optimal stopping and American options 187 8.2 Implied volatility 188 8.3 Alternatives to and modifications of the Black-Scholes model and pricing theory 192 8.3.1 The implied volatility function (IVF) model 192 8.3.2 The constant elasticity of variance (CEV) model 192 8.3.3 The stochastic volatility (SV) model 193 8.3.4 Nonparametric methods 194 8.3.5 A combined substantive-empirical approach 195 Exercises 197 Advanced Multivariate and Time Series Methods in Financial Econometrics 199 9.1 Canonical correlation analysis 200 9.1.1 Cross-covariance and correlation matrices 200 9.1.2 Canonical correlations 201 9.2 Multivariate regression analysis 203 9.2.1 Least squares estimates in multivariate regression 203 9.2.2 Reduced-rank regression 203 9.3 Modified Cholesky decomposition and high-dimensional covariance matrices 205 9.4 Multivariate time series 206 9.4.1 Stationarity and cross-correlation 206 9.4.2 Dimension reduction via PCA 206 9.4.3 Linear regression with stochastic regressors 207 9.4.4 Unit-root tests 211 9.4.5 Cointegrated VAR 213 9.5 Long-memory models and regime switching/structural change 217 9.5.1 Long memory in integrated models 217 9.5.2 Change-point AR-GARCH models 219 9.5.3 Regime-switching models 224 9.6 Stochastic volatility and multivariate volatility models 225 9.6.1 Stochastic volatility models 225 9.6.2 Multivariate volatility models 228 9.7 Generalized method of moments (GMM) 229 9.7.1 Instrumental variables for linear relationships 229 9.7.2 Generalized moment restrictions and GMM estimation 231
xviii Contents 9.7.3 An example: Comparison of different short-term interest rate models 233 Exercises 234 10 Interest Rate Markets 239 10.1 Elements of interest rate markets -.: 240 10.1.1 Bank account (money market account) and short rates 241 10.1.2 Zero-coupon bonds and spot rates 241 10.1.3 Forward rates 244 10.1.4 Swap rates and interest rate swaps 245 ^10.1.5 Caps, floors, and swaptions 247 10.2 Yield curve estimation 247 10.2.1 Nonparametric regression using spline basis functions 248 10.2.2 Parametric models 248 10.3 Multivariate time series of bond yields and other interest rates 252 10.4 Stochastic interest rates and short-rate models 255 10.4.1 Vasicek, Cox-Ingersoll-Ross, and Hull-White models... 258 10.4.2 Bond option prices 259 10.4.3 Black-Karasinski model 260 10.4.4 Multifactor affine yield models 261 10.5 Stochastic forward rate dynamics and pricing of LIBOR and swap rate derivatives 261 10.5.1 Standard market formulas based on Black's model of forward prices 262 10.5.2 Arbitrage-free pricing: martingales and numeraires 263 10.5.3 LIBOR and swap market models 264 10.5.4 The HJM models of the instantaneous forward rate... 266 10.6 Parameter estimation and model selection 267 10.6.1 Calibrating interest rate models in the financial industry 267 10.6.2 Econometric approach to fitting term-structure models 270 10.6.3 Volatility smiles and a substantive-empirical approach.. 271 Exercises 272 11 Statistical Trading Strategies 275 11.1 Technical analysis, trading strategies, and data-snooping checks 277 11.1.1 Technical analysis 277 11.1.2 Momentum and contrarian strategies 279
11.1.3 Pairs trading strategies 279 11.1.4 Empirical testing of the profitability of trading strategies 282 11.1.5 Value investing and knowledge-based trading strategies 285 11.2 High-frequency data, market microstructure, and associated trading strategies 286 11.2.1 Institutional background and stylized facts about transaction data 287 11.2.2 Bid-ask bounce and nonsynchronous trading models... 291 11.2.3 Modeling time intervals between trades 292 11.2.4 Inference on underlying price process 297 11.2.5 Real-time trading systems 299 11.3 Transaction costs and dynamic trading 300 11.3.1 Estimation and analysis of transaction costs 300 11.3.2 Heterogeneous trading objectives and strategies 300 11.3.3 Multiperiod trading and dynamic strategies 301 Exercises 302 12 Statistical Methods in Risk Management 305 12.1 Financial risks and measures of market risk 306 12.1.1 Types of financial risks 306 12.1.2 Internal models for capital requirements 307 12.1.3 VaR and other measures of market risk 307 12.2 Statistical models for VaR and ES 309 12.2.1 The Gaussian convention and the ^-modification 309 12.2.2 Applications of PCA and an example 310 12.2.3 Time series models 311 12.2.4 Backtesting VaR models 311 12.3 Measuring risk for nonlinear portfolios 312 12.3.1 Local valuation via Taylor expansions 312 12.3.2 Full valuation via Monte Carlo 314 12.3.3 Multivariate copula functions 314 12.3.4 Variance reduction techniques 316 12.4 Stress testing and extreme value theory 318 12.4.1 Stress testing 318 12.4.2 Extraordinary losses and extreme value theory 318 12.4.3 Scenario analysis and Monte Carlo simulations 321 Exercises 321 xix
xx Contents Appendix A. Martingale Theory and Central Limit Theorems. 325 Appendix B. Limit Theorems for Stationary Processes 331 Appendix C. Limit Theorems Underlying Unit-Root Tests and Cointegration 333 References 337 Index 349