statistics and finance - content

8/3/2019 Statistics and Finance - Content

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David Ruppert

Statistics and FinanceAn Introduction

Springer


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Contents

Notation xxi1 Introduction 1

1.1 References 52 Probability and Statistical Models 7

2.1 Introduction 72.2 Axioms of Probability 72.2.1 Independence 8

2.2.2 Bayes' law 82.3 Probability Distributions 9

2.3.1 Random variables 92.3.2 Independence 102.3.3 Cumulative distribution functions 102.3.4 Quantiles and percentiles 112.3.5 Expectations and variances 122.3.6 Does the expected value exist? 132.4 Functions of Random Variables 14

2.5 Random Samples 152.6 The Binomial Distribution 162.7 Location, Scale, and Shape Parameters 172.8 Some Common Continuous Distributions 17

2.8.1 Uniform distributions 172.8.2 Normal distributions 182.8.3 The lognormal distribution 202.8.4 Exponential and double exponential distributions 21

2.9 Sampling a Normal Distribution 212.9.1 Chi-squared distributions 212.9.2 t-distributions 222.9.3 ^-distributions 22

2.10 Order Statistics and the Sample CDF 23


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Contents2.10.1 Normal probability plots 23

2.11 Skewness and Kurtosis 242.12 Heavy-Tailed Distributions 28

2.12.1 Double exponential distributions 302.12.2 t-distributions have heavy tails 302.12.3 Mixture models 312.12.4 Pareto distributions 322.12.5 Distributions with Pareto tails 34

2.13 Law of Large Numbers and Central Limit Theorem 362.14 Multivariate Distributions 37

2.14.1 Correlation and covariance 382.14.2 Independence and covariance 402.14.3 The multivariate normal distribution 412.15 Prediction 422.15.1 Best linear prediction 422.15.2 Prediction error in linear prediction 432.15.3 Multivariate linear prediction 44

2.16 Conditional Distributions 442.16.1 Best prediction 452.16.2 Normal distributions: Conditional expectations and

variance 452.17 Linear Functions of Random Variables 462.17.1 Two linear combinations of random variables 48

2.17.2 Independence and variances of sums 492.17.3 Application to normal distributions 49

2.18 Estimation 492.18.1 Maximum likelihood estimation 502.18.2 Standard errors 522.18.3 Fisher information 532.18.4 Bayes estimation* 542.18.5 Robust estimation* 56

2.19 Confidence Intervals 602.19.1 Confidence interval for the mean 602.19.2 Confidence intervals for the variance and standard

deviation 612.19.3 Confidence intervals based on standard errors 62

2.20 Hypothesis Testing 622.20.1 Hypotheses, types of errors, and rejection regions 622.20.2 P-values 632.20.3 Two-sample i-tests 642.20.4 Statistical versus practical significance 652.20.5 Tests of normality 662.20.6 Likelihood ratio tests 66

2.21 Summary 682.22 Bibliographic Notes 70


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Contents xiii2.23 References 712.24 Problems 72Returns 753.1 Introduction 75

3.1.1 Net returns 753.1.2 Gross returns 753.1.3 Log returns 763.1.4 Adjustment for dividends 77

3.2 Behavior Of Returns 783.3 The Random Walk Model 80

3.3.1 I.i.d. normal returns 803.3.2 The lognormal model 803.3.3 Random walks 823.3.4 Geometric random walks 823.3.5 The effect of the drift \x 833.3.6 Are log returns normally distributed? 843.3.7 Do the GE daily returns look like a geometric random

walk? 863.4 Origins of the Random Walk Hypothesis 89

3.4.1 Fundamental analysis 893.4.2 Technical analysis 91

3.5 Efficient Markets Hypothesis (EMH) 933.5.1 Three types of efficiency 933.5.2 Testing market efficiency 94

3.6 Discrete and Continuous Compounding 953.7 Summary 963.8 Bibliographic Notes 973.9 References 973.10 Problems 98

Time Series Models 1014.1 Time Series Data 1014.2 Stationary Processes 102

4.2.1 Weak white noise 1034.2.2 Predicting white noise 1044.2.3 Estimating parameters of a stationary process 104

4.3 AR(1) Processes 1054.3.1 Properties of a stationary AR(1) process 1064.3.2 Convergence to the stationary distribution 1084.3.3 Nonstationary AR(1) processes 1084.4 Estimation of AR(1) Processes 1104.4.1 Residuals and model checking I l l4.4.2 AR(1) model for GE daily log returns 113

4.5 AR(p) Models 115


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xiv Contents4.5.1 AR(6) model for GE daily log retu rns 1174.6 Moving Average (MA) Processes 1184.6.1 MA(1) processes 1184.6.2 General MA processes 1184.6.3 MA(2) model for GE daily log retu rns 1194.7 ARIMA Processes 1204.7.1 The backwards operator 1204.7.2 ARM A processes 1204.7.3 Fit ting ARMA processes: GE daily log retu rns 1214.7.4 Th e differencing operator 1224.7.5 From ARMA processes to ARIMA processes 1224.7.6 ARIM A(2,l,0) model for GE daily log prices 1234.8 Model Selection 1244.8.1 AIC and SBC 1244.8.2 GE daily log return s: Choosing the AR order 1254.9 Three-M onth Treasury Bill Ra tes 1264.10 Forecasting 1284.10.1 Forecasting GE daily log retu rns and log prices 1304.11 Summary 1314.12 Bibliographic Notes 1344.13 References 134

4.14 Problems 1345 Portfolio Th eory .' 1375.1 Trading Off Expected Return and Risk 1375.2 One Risky Asset and One Risk-Free Asset 1375.2.1 Estimating E(R) and CTR 1405.3 Two Risky Assets 1405.3.1 Risk versus expected retu rn 1405.3.2 Estim ating means, standard deviations, and covariancesl41

5.4 Combining Two Risky Assets with a Risk-Free Asset 1425.4.1 Tangency portfolio with two risky assets 1425.4.2 Com bining the tangency portfolio with the risk-freeasset 1445.4.3 Effect of p12 1465.5 Risk-Efficient Portfolios with N Risky Assets* 1465.5.1 Efficient-portfolio mathem atics 1465.5.2 The minimum variance portfolio 1525.5.3 Selling short 1545.5.4 Back to the math Finding the tangency po rtfo lio ... 1565.5.5 Examples 1575.6 Quadratic Programm ing* 1605.7 Is the Theory Useful? 1635.8 Utility Theory* 1645.9 Sum mary 165


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Contents xv5.10 Bibliographic Notes 1665.11 References 1665.12 Problem s 166Regression 1696.1 Introduction 1696.1.1 Straight line regression 1706.2. Least Squares Estim ation 1706.2.1 Estim ation in straight line regression 1716.2.2 Variance of ft '. 1726.2.3 Estim ation in multiple linear regression 1746.3 Standard Errors, T-Values, and P-Values 1746.4 Analysis Of Variance, R2, and F-T ests 1776.4.1 AOV tab le 1776.4.2 Sums of squares (SS) and R2 1776.4.3 Degrees of freedom (DF) 1786.4.4 Mean sums of squares (MS) and testing 1786.4.5 Adjusted R2 1796.4.6 Sequential and partia l sums of squares 1806.5 Regression Hedging* 1816.6 Regression and Best Linear Prediction 1836.7 Model Selection 1836.8 Collinearity and Variance Inflation 1876.9 Centering the Pred ictors 1896.10 Nonlinear Regression 1896.11 The General Regression Model 1926.12 Troubleshooting 1936.12.1 Influence diagnostics and residuals 1946.12.2 Residual analysis 1996.13 Transform-Both-Sides Regression* 2066.13.1 How TBS works 210

6.13.2 Power transform ations 2146.14 The Geom etry of Transformations* 2146.15 Robust Regression* 2166.16 Summary 2176.17 Bibliographic Notes 2196.18 References 2206.19 Problems 220The Capital Asset Pricing Model 2257.1 Introduction to CAPM 2257.2 The Capital Market Line (CML) 2277.3 Betas and the Security Market Line : 2307.3.1 Exam ples of beta s 2317.3.2 Comparison of the CML with the SML 231


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xvi Contents7.4 The Security Characteristic Line 2327.4.1 Reducing unique risk by diversification 2347.4.2 Can beta be negative? 2357.4.3 Are the assumptions sensible? 2357.5 Some More Portfolio Theory 2367.5.1 Contributions to the market portfolio's risk 2367.5.2 Derivation of the SML 2377.6 Estimation of Beta and Testing the CAPM 2387.6.1 Regression using returns instead of excess returns 2417.6.2 Interpretation of alpha 2417.7 Using CAPM in Portfolio Analysis 2427.8 Factor Models 2427.8.1 Estimating expectations and covariances of assetreturns 2447.8.2 Fama and French three-factor model 2457.8.3 Cross-sectional factor models 2467.9 An Interesting Question* 2467.10 Is Beta Constant?* 2507.11 Summary 2527.12 Bibliographic Notes 2547.13 References 2547.14 Problems 255

8 Options Pricing 2578.1 Introduction 2578.2 Call Options 2588.3 The Law of One Price 2598.3.1 Arbitrage 2598.4 Time Value of Money and Present Value 2608.5 Pricing Calls A Simple Binomial Example 2608.6 Two-Step Binomial Option Pricing 2638.7 Arbitrage Pricing by Expectation 2648.8 A General Binomial Tree Model 2668.9 Martingales 2688.9.1 Martingale or risk-neutral measure 2698.9.2 The risk-neutral world 2698.10 From Trees to Random Walks and Brownian Motion 2708.10.1 Getting more realistic 270

8.10.2 A three-step binomial tree 2708.10.3 More time steps 2718.10.4 Properties of Brownian motion 2738.11 Geometric Brownian Motion 2738.12 Using the Black-Scholes Formula 2768.12.1 How does the option price depend on the inputs? 2768.12.2 Early exercise of calls is never optimal 277


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Contents xvii8.12.3 Are there retu rns on non trading days? 2788.13 Implied Volatility 2798.13.1 Volatility smiles and polynomial regression* 2808.14 Put s 2848.14.1 Pricing pu ts by binomial trees 2848.14.2 Why are puts different from calls? 2878.14.3 Put-call parity 2878.15 The Evolution of Option Prices ; 2888.16 Leverage of Options and Hedging 2898.17 The Greeks 2908.17.1 Delta and Gam ma hedging 2948.18 Intrins ic Value and Tim e Value* : 2958.19 Sum mary 295

8.20 Bibliographic Notes 2978.21 References 2988.22 Problems 2989 Fixed Incom e Securities 3019.1 Introduction 3019.2 Zero-Coupon Bonds 3029.2.1 Price and retu rns fluctuate with the interest rate 3029.3 Coupon Bonds 304

9.3.1 A general formula 3059.4 Yield to Maturity 3059.4.1 General method for yield to maturity 3069.4.2 MATLAB functions 3079.4.3 Spot rates 3089.5 Term Struc ture 3099.5.1 Introduction: Interest rates depend upon m aturity . . . . 3099.5.2 Describing the term structure 3109.6 Continuous Com pounding 3149.7 Continuous Forward Rates 3159.8 Sensitivity of Price to Yield 3169.8.1 Duration of a coupon bond 3179.9 Estim ation of a Continuous Forward Rate* 3179.10 Sum mary 3229.11 Bibliographic Notes 3239.12 References 3249.13 Prob lems 32410 Resampling 32710.1 Introduction 32710.2 Confidence Intervals for the Mean 32810.3 Resam pling and Efficient Portfolios 33210.3.1 The global asset allocation problem 332


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xviii Contents10.3.2 Uncertain ty about mean-variance efficient p ort fo lios. .. 33410.3.3 W hat if we knew the expected returns? 33710.3.4 W hat if we knew th e covariance matrix? 33810.3.5 W hat if we had more da ta? 33910.4 Bagging* 34010.5 Summary 34210.6 Bibliographic Notes 34310.7 References 34310.8 Problem s 343

11 Value-At-Risk 34511.1 The Need for Risk Management 34511.2 VaR with One Asset 34611.2.1 Non parametric estimation of VaR 34611.2.2 Parametric estimation of VaR 34811.2.3 Estim ation of VaR assuming Pa reto tails* 34911.2.4 Estim ating the tail index* 35011.2.5 Confidence intervals for VaR using resam pling 35311.2.6 VaR for a derivative 35411.3 VaR for a Portfolio of Assets 35511.3.1 Portfolios of stocks only 35511.3.2 Portfolios of one stock and an option on th at stock . . . 35611.3.3 Portfolios of one stock and an option on ano ther stock 35611.4 Choosing the Holding Period and Confidence 35711.5 VaR and Risk Management 35711.6 Summary 35911.7 Bibliographic Notes 36011.8 References 36011.9 Problems 360

12 GARCH Models 36312.1 Introd uction : 36312.2 Modeling Conditional Means and Variances 36412.3 ARC H(l) Processes 36512.4 Th e AR (1)/ARC H(1) Model 36812.5 ARCH(g) Models 37012.6 GARCH(p, q) Models 37012.7 GARCH Processes Have Heavy Tails 37112.8 Comparison of ARMA and GARCH Processes 37212.9 Fitt ing GARCH Models 37212.10 I-GARCH Models .37 712.10.1 W ha t does it mean to have an infinite variance? 37912.11 GARCH-M Processes 38112.12 E-GARCH 38312.13 The GARCH Zoo* 386


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Contents xix12.14 Applications of GARCH in Finance 38612.15 Pricing Options Under Generalized GARCH Processes* 38712.16 Sum mary 39112.17 Bibliographic Notes 39212.18 References 39212.19 Problems ; 393

13 Nonparametric Regression and Splines 39713.1 Introduction 39713.2 Choosing a Regression Method 40013.2.1 Nonparam etric regression 40013.2.2 Linear 40013.2.3 Nonlinear param etric regression 40013.2.4 Com parison of linear and nonparametric regression . . . 40113.3 Linear Splines 40513.3.1 Linear splines with one kno t 40513.3.2 Linear splines with many kno ts 40613.4 Other Degree Splines 40713.4.1 Quadra tic splines 40713.4.2 pth degree splines 40813.5 Least Squares Estim ation 40913.6 Selecting the Spline Param eters 41013.6.1 Es timating the volatility function 41313.7 Additive Models* 41513.8 Penalized Splines* 41813.8.1 Penalizing the jumps at the knots 42013.8.2 Cross-validation 42113.8.3 The effective number of param eters 42313.8.4 Generalized cross-validation 42513.8.5 AIC 42613.8.6 Penalized splines in MATLAB 427

13.9 Summary 43113.10 Bibliographic Notes 43113.11 References 43113.12 Problems 43214 Behavioral Finance 43514.1 Introduction 43514.2 Defense of the EMH 43614.3 Challenges to the EMH 43714.4 Can Arbitrageurs Save the Day? 43814.5 W hat Do the D ata Say? 43914.5.1 Excess price volatility 43914.5.2 The overreaction hypothesis 43914.5.3 Reactions to earnings announcem ents 440


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xx Contents14.5.4 Counter-argum ents to pricing anomalies 44114.5.5 Reaction to non-news 44214.6 Market Volatility and Irrational Exuberance 44314.6.1 Best prediction 444

14.7 Th e Current Statu s of Classical Finance 44514.8 Bibliographic Notes 44514.9 References 44614.10 Problem s 447Glossary 449Index 461

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