Statistical Models and Methods for Financial Markets

Transcription

1 Tze Leung Lai/ Haipeng Xing Statistical Models and Methods for Financial Markets B Q Springer

2 Preface \ vii Part I Basic Statistical Methods and Financial Applications 1 Linear Regression Models Ordinary least squares (OLS) Residuals and their sum of squares Properties of projection matrices Properties of nonnegative definite matrices Statistical properties of OLS estimates Statistical inference Confidence intervals ANOVA (analysis of variance) tests Variable selection Test-based and other variable selection criteria Stepwise variable selection Regression diagnostics Analysis of residuals Influence diagnostics Extension to stochastic regressors Minimum-variance linear predictors Futures markets and hedging with futures contracts Inference in the case of stochastic regressors Bootstrapping in regression " The plug-in principle and bootstrap resampling Bootstrapping regression models Bootstrap confidence intervals Generalized least squares 25

3 1.8 Implementation and illustration 26 Exercises 32 Multivariate Analysis and Likelihood Inference Joint distribution of random variables Change of variables Mean and covariance matrix Principal component analysis (PCA) Basic definitions Properties of principal components An example: PCA of U.S. Treasury-LIBOR swap rates Multivariate normal distribution Definition and density function Marginal and conditional distributions Orthogonality and independence, with applications to regression Sample covariance matrix and Wishart distribution... 52, 2.4 Likelihood inference Method of maximum likelihood Asymptotic inference Parametric bootstrap 59 Exercises 60 Basic Investment Models and Their Statistical Analysis Asset returns Definitions Statistical models for asset prices and returns Markowitz's portfolio theory Portfolio weights Geometry of efficient sets Computation of efficient portfolios Estimation of ft and S and an example Capital asset pricing model (CAPM) The model Investment implications ; Estimation and testing Empirical studies of CAPM Multifactor pricing models Arbitrage pricing theory Factor analysis The PCA approach The Fama-French three-factor model 86

4 xv 3.5 Applications of resampling to portfolio management Michaud's resampled efficient frontier Bootstrap estimates of performance 88 Exercises 89 Parametric Models and Bayesian Methods Maximum likelihood and generalized linear models Numerical methods for computing MLE Generalized linear models Nonlinear regression models The Gauss-Newton algorithm Statistical inference Implementation and an example Bayesian inference Prior and posterior distributions Bayes procedures Bayes estimators of multivariate normal mean and covariance matrix Bayes estimators in Gaussian regression models Empirical Bayes and shrinkage estimators Investment applications of shrinkage estimators and Bayesian methods Shrinkage estimators of fi and for the plug-in efficient frontier An alternative Bayesian approach Ill Exercises 113 Time Series Modeling and Forecasting Stationary time series analysis Weak stationarity Tests of independence Wold decomposition and MA, AR, and ARMA models Forecasting in ARMA models Parameter estimation and order determination Analysis of nonstationary time series Detrending ' An empirical example Transformation and differencing Unit-root nonstationarity and ARIMA models Linear state-space models and Kalman filtering Recursive formulas for P t t _i,x t t _i, and x t t Dynamic linear models and time-varying betas in CAPM 133

5 xvi Contents Exercises Dynamic Models of Asser Returns and Their Volatilities Stylized facts on time series of asset returns Moving average estimators of time-varying volatilities Conditional heteroskedastic models The ARCH model The GARCH model The integrated GARCH model The exponential GARCH model The ARMA-GARCH and ARMA-EGARCH models Forecasting future returns and volatilities Implementation and illustration 156 Exercises 157 Part II Advanced Topics in Quantitative Finance 7 Nonparametric Regression and Substantive-Empirical Modeling Regression functions and minimum-variance prediction Univariate predictors Running-mean/running-line smoothers and local polynomial regression Kernel smoothers Regression splines Smoothing cubic splines Selection of smoothing parameter The bias-variance trade-off Cross-validation Multivariate predictors Tensor product basis and multivariate adaptive regression splines Additive regression models Projection pursuit regression Neural networks A modeling approach that combines domain knowledge with nonparametric regression Penalized spline models and estimation of forward rates A semiparametric penalized spline model for the forward rate curve of corporate debt 178 Exercises 179

6 xvii Option Pricing and Market Data Option prices and pricing theory Options data and put-call parity The Black-Scholes formulas for European options Optimal stopping and American options Implied volatility Alternatives to and modifications of the Black-Scholes model and pricing theory The implied volatility function (IVF) model The constant elasticity of variance (CEV) model The stochastic volatility (SV) model Nonparametric methods A combined substantive-empirical approach 195 Exercises 197 Advanced Multivariate and Time Series Methods in Financial Econometrics Canonical correlation analysis Cross-covariance and correlation matrices Canonical correlations Multivariate regression analysis Least squares estimates in multivariate regression Reduced-rank regression Modified Cholesky decomposition and high-dimensional covariance matrices Multivariate time series Stationarity and cross-correlation Dimension reduction via PCA Linear regression with stochastic regressors Unit-root tests Cointegrated VAR Long-memory models and regime switching/structural change Long memory in integrated models Change-point AR-GARCH models Regime-switching models Stochastic volatility and multivariate volatility models Stochastic volatility models Multivariate volatility models Generalized method of moments (GMM) Instrumental variables for linear relationships Generalized moment restrictions and GMM estimation 231

7 xviii Contents An example: Comparison of different short-term interest rate models 233 Exercises Interest Rate Markets Elements of interest rate markets -.: Bank account (money market account) and short rates Zero-coupon bonds and spot rates Forward rates Swap rates and interest rate swaps 245 ^ Caps, floors, and swaptions Yield curve estimation Nonparametric regression using spline basis functions Parametric models Multivariate time series of bond yields and other interest rates Stochastic interest rates and short-rate models Vasicek, Cox-Ingersoll-Ross, and Hull-White models Bond option prices Black-Karasinski model Multifactor affine yield models Stochastic forward rate dynamics and pricing of LIBOR and swap rate derivatives Standard market formulas based on Black's model of forward prices Arbitrage-free pricing: martingales and numeraires LIBOR and swap market models The HJM models of the instantaneous forward rate Parameter estimation and model selection Calibrating interest rate models in the financial industry Econometric approach to fitting term-structure models Volatility smiles and a substantive-empirical approach Exercises Statistical Trading Strategies Technical analysis, trading strategies, and data-snooping checks Technical analysis Momentum and contrarian strategies 279

8 Pairs trading strategies Empirical testing of the profitability of trading strategies Value investing and knowledge-based trading strategies High-frequency data, market microstructure, and associated trading strategies Institutional background and stylized facts about transaction data Bid-ask bounce and nonsynchronous trading models Modeling time intervals between trades Inference on underlying price process Real-time trading systems Transaction costs and dynamic trading Estimation and analysis of transaction costs Heterogeneous trading objectives and strategies Multiperiod trading and dynamic strategies 301 Exercises Statistical Methods in Risk Management Financial risks and measures of market risk Types of financial risks Internal models for capital requirements VaR and other measures of market risk Statistical models for VaR and ES The Gaussian convention and the ^-modification Applications of PCA and an example Time series models Backtesting VaR models Measuring risk for nonlinear portfolios Local valuation via Taylor expansions Full valuation via Monte Carlo Multivariate copula functions Variance reduction techniques Stress testing and extreme value theory Stress testing Extraordinary losses and extreme value theory Scenario analysis and Monte Carlo simulations 321 Exercises 321 xix

9 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