Financial Econometrics, Mathematicsand Statistics

Cheng-Few Lee • Hong-Yi Chen •

John Lee

Financial Econometrics,Mathematicsand StatisticsTheory, Method and Application

123

Cheng-Few LeeDepartment of Finance and EconomicsRutgers Business SchoolRutgers UniversityPiscataway, NJ, USA

Hong-Yi ChenDepartment of FinanceNational Chengchi UniversityTaipei, Taiwan

John LeeCenter for PBBEF ResearchMorris Plains, NJ, USA

ISBN 978-1-4939-9427-4 ISBN 978-1-4939-9429-8 (eBook)https://doi.org/10.1007/978-1-4939-9429-8

Library of Congress Control Number: 2019932616

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Preface

We draw upon our years of teaching, research, and practice on the subjects offinancial econometrics, mathematics and statistics for this textbook. Overall,our goal is to provide an advanced level book that reviews, discusses, andintegrates financial econometrics, mathematics and statistics.

We focus on five principles to frame our presentation of this book:

(1) To discuss the basic methodology of financial econometrics, mathe-matics and statistics,

(2) To show how econometric methodologies can be used in finance andaccounting-related research, which includes single equation, multipleregression, simultaneous regression, panel data analysis, time-seriesanalysis, spectral analysis, nonparametric analysis, semiparametricanalysis, GMM analysis, and other methods,

(3) To show how financial mathematics such as Itô’s calculus is importantto derive the intertemporal capital asset pricing model and optionpricing model,

(4) To demonstrate how statistics distribution, such as normal distribution,stable distribution, and lognormal distribution, has been used in researchrelated to portfolio theory and risk management,

(5) To show how binomial distribution, lognormal distribution, noncentralchi-square distribution, Poisson distribution, and others have been usedin studies related to option and futures.

In order to comprehend this book, the reader needs two semesters ofeconometrics, two semesters of mathematical statistics, and one semester ofmultivariate statistics.

We divide this book into four parts: Regression and Financial Econo-metrics; Time-Series Analysis; Statistical Distributions, Option PricingModel, and Risk Management; and Statistics, Itô’s Calculus, and OptionPricing Model.

PART I: Regression and Financial Econometrics

There are seven chapters in this part. In Chap. 2, we discuss the assumptionsof the multiple regression model, estimated parameters of the multipleregression model, the standard error of the residual estimate, and thecoefficient of determination. We also investigate tests on sets and individual

v

regression coefficients and the confidence interval for the mean response andprediction interval for the individual response.

In Chap. 3, we discuss various topics associated with the regressionanalysis, including multicollinearity, heteroscedasticity, autocorrelation,model specification and specification bias of the regression model, nonlinearregression models, lagged dependent variables in the regression model,dummy variables in the regression model, and regression model with inter-action variables. We also apply the regression approach to investigate theeffect of alternative business strategies and apply the logistic regressionmodel to credit risk analysis.

In Chap. 4, we extend single-equation models to simultaneous equationmodels. Specifically, we discuss simultaneous equation system, two-stageleast squares method, and three-stage least squares method. In Chap. 5, wediscuss an econometric approach to financial analysis, planning, and fore-casting. The issue of simultaneity and the dynamics of corporate-budgetingdecisions will be explored by using finance theory. We also investigate theinterrelationships among the programming, the simultaneous equations, andthe econometrics approaches.

Chapter 6 addresses one of the important issues related to panel dataanalysis. We introduce the dummy variable technique and the error com-ponent model for analyzing pooled data. We investigate the possible impactsof firm effect and time effect on choosing the optimal functional form of afinancial research study. In this chapter, we also discuss the criteria of usingfixed effects or random effects approach.

Chapter 7 discusses how errors-in-variables estimation methods are usedin finance research. We show how errors-in-variables problems can affect theestimators of the linear regression model, as well as discuss the effects theyhave on the empirical research cost of capital, asset pricing, capital structure,and investment decision. Chapter 8 provides three alternativeerrors-in-variables estimation models in testing the capital asset pricingmodel. Specifically, we present three alternative correction methods for theerrors-in-variables problem. In Chap. 9, we discuss the issue of spuriousregression and data mining in both conditional asset pricing models andsimple predictive regression. We also discuss the impact of spuriousregression and data mining on conditional asset pricing.

PART II: Time-Series Analysis and Its Applications

The purpose of Chap. 10 is to describe the components of time-seriesanalyses and to discuss alternative methods of economic and business fore-casting in terms of time-series data. Specifically, we discuss a classicaldescription of three time-series components, the moving-average and sea-sonally adjusted time series, linear and log-linear time trend regressions,exponential smoothing and forecasting, autoregressive forecasting model,ARIMA model, and composite forecasting.

In Chap. 11, we attempt to achieve two goals. First, we present alternativetheories for deriving optimal hedge ratios. We discuss various estimationmethods and the relationship among lengths of hedging horizon, maturity of

vi Preface

futures contract, data frequency, and hedging effectiveness. Second, we showhow SAS program can be used to estimate hedge ratio in terms of ARCHmethod, GARCH method, EGARCH method, GJR-GARCH method, andTGARCH method.

PART III: Statistical Distributions, Option Pricing Model and RiskManagement

Statistical distributions such as binomial distribution, multinomial distribu-tion, normal distribution, lognormal distribution, Poisson distribution, centralchi-square distribution, noncentral chi-square distribution, copula distribu-tion, nonparametric distribution, and other distributions are important infinance research. In Chap. 12, we will discuss how binomial and multinomialdistribution can be used to derive the option pricing model. In Chap. 13, weshow how to use two alternative binomial option pricing model approachesto derive Black–Scholes option pricing model.

In Chap. 14, we will discuss how normal and lognormal distribution canbe used to derive the option pricing model. In Chap. 15, we will show howcopula distribution can be used to do credit risk analysis. In Chap. 16, we willshow how multivariate analyses such as factor analysis and discriminantanalysis can be used to do financial rating analysis. In Part IV, we willcontinue to discuss how statistics distribution can be used to derive optionpricing model. In addition, we will also show how Itô’s calculus can be usedto derive option pricing model.

PART IV: Statistics, Itô’s Calculus, and Option Pricing Model

In Chap. 17, we will show how characteristic function and noncentralchi-square can be used to analyze stochastic volatility option pricing model.In Chap. 18, we will discuss alternative methods to estimate implied vari-ance. In Chap. 19, we will show the numerical valuation of Asian optionswith higher moments in the underlying distribution. Both European andAmerican options will be discussed in this chapter. In Chap. 20, we will firstreview Itô Lemma and stochastic differential equation, and then we will showhow this mathematical technique can be used to derive option pricing model.In Chap. 21, we will discuss the relationship between binomial option pricingmodel and Black–Scholes option pricing model. In addition, we also showhow to use stochastic calculus to derive Black–Scholes model in detail. InChap. 22, we will show how to use noncentral chi-square distribution toderive constant elasticity of variance option pricing model. In Chap. 23, wewill discuss option pricing and hedging performance under stochasticvolatility and stochastic interest rates. Finally, in Chap. 24, we will show hownonparametric distribution can be used to derive option bounds. Someempirical studies or option bounds are also provided.

This textbook can be used for the quantitative finance program and Ph.D.programs in economics, statistics, and finance. It demonstrates how to applydifferent econometrics and statistical methods in finance research. In

Preface vii

addition, applications of Itô’s calculus in deriving option pricing model arealso discussed in some detail.

It is well known that financial econometrics, mathematics and statistics arethree of the most important tools to solve theoretical and practical issues ofquantitative finance. This textbook uses real-world data to show how thesethree quantitative tools can be used to solve quantitative finance issues inasset pricing, option pricing models, risk management, and other quantitativefinance issues. This textbook uses a traditional approach by combiningresearch papers from journals, handbooks, and textbooks to streamline thesetopics. We take advantage of using our own research papers, edited hand-book, and textbook to formulate a meaningful, unique, comprehensivetextbook. We heavily draw upon Handbook of Quantitative Finance and RiskManagement (Springer 2009), Statistics for Business and Financial Eco-nomics, 3rd ed. (Springer 2013), and Essentials of Excel, Excel VBA, SASand Minitab for Statistical and Financial Analyses (Springer 2016), as wellas Review of Quantitative Finance and Accounting and other journals withwhich we have published relevant papers.

Note that this textbook is intended to be used in its entirety instead ofchapter by chapter. Readers may find the aforementioned Springer volumes’useful references during the learning process. For example, there are severalchapters wherein we draw from Handbook of Quantitative Finance and RiskManagement—here, the reader can refer to the handbook. Similarly, Chaps.2 and 3 are expanded versions of Statistics for Business and FinancialEconomics, 3rd ed.

There are undoubtedly some errors in the finished product such as con-ceptual, grammatical, or methodological. We would like to invite readers tosend their suggestions, comments, criticisms, and corrections to the author,Professor Cheng F. Lee at the Department of Finance and Economics, Rut-gers University at 100 Rockafeller Road, Room 5188, Piscataway, NJ 08854.Alternatively, readers can send this information by email to either Cheng FewLee (cflee@business.rutgers.edu) or Hong-Yi Chen (fnhchen@nccu.edu.tw).

Piscataway, USA Cheng-Few LeeTaipei, Taiwan Hong-Yi ChenMorris Plains, USA John LeeMay 2019

viii Preface

Contents

1 Introduction to Financial Econometrics, Mathematics,and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Regression and Financial Econometrics . . . . . . . . . . . . . 2

1.2.1 Single-Equation Regression Methods . . . . . . . . 21.2.2 Simultaneous Equation Models . . . . . . . . . . . . 41.2.3 Panel Data Analysis . . . . . . . . . . . . . . . . . . . . . 41.2.4 Alternative Methods to Deal

with Measurement Error . . . . . . . . . . . . . . . . . 41.2.5 Time-Series Analysis . . . . . . . . . . . . . . . . . . . . 5

1.3 Financial Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.1 Statistical Distributions . . . . . . . . . . . . . . . . . . 51.3.2 Principle Components and Factor Analysis . . . 61.3.3 Nonparametric and

Semiparametric Analyses . . . . . . . . . . . . . . . . . 61.3.4 Cluster Analysis. . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Applications of Financial Econometrics, Mathematicsand Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.1 Asset Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.2 Corporate Finance . . . . . . . . . . . . . . . . . . . . . . 61.4.3 Financial Institution . . . . . . . . . . . . . . . . . . . . . 71.4.4 Investment and Portfolio Management . . . . . . . 71.4.5 Option Pricing Model . . . . . . . . . . . . . . . . . . . 71.4.6 Futures and Hedging . . . . . . . . . . . . . . . . . . . . 71.4.7 Mutual Fund . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4.8 Credit Risk Modeling. . . . . . . . . . . . . . . . . . . . 71.4.9 Other Applications . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Overall Discussion of This Book . . . . . . . . . . . . . . . . . . 81.5.1 Regression and Financial Econometrics . . . . . . 81.5.2 Time-Series Analysis and Its Application . . . . 91.5.3 Statistical Distributions and Option Pricing

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5.4 Statistics, Itô’s Calculus and Option Pricing

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

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1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Appendix: Keywords for Chaps. 2–24 . . . . . . . . . . . . . . . . . . . . 11Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Part I Regression and Financial Econometrics

2 Multiple Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 The Model and Its Assumptions . . . . . . . . . . . . . . . . . . . 202.3 Estimating Multiple Regression Parameters. . . . . . . . . . . 232.4 The Residual Standard Error and the Coefficient

of Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5 Tests on Sets and Individual Regression Coefficients . . . 262.6 Confidence Interval for the Mean Response

and Prediction Interval for the Individual Response . . . . 302.7 Business and Economic Applications . . . . . . . . . . . . . . . 332.8 Using Computer Programs to Do Multiple Regression

Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.8.1 SAS Program for Multiple Regression

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Appendix 1: Derivation of the Sampling Varianceof the Least Squares Slope Estimations . . . . . . . . . . . . . . . . . . . . 48Appendix 2: Cross-sectional Relationship Among Price PerShare, Dividend Per Share, and Return Earning Per Share . . . . . . . . 49Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3 Other Topics in Applied Regression Analysis . . . . . . . . . . . . . 553.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2 Multicollinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.3 Heteroscedasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.4 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.5 Model Specification and Specification Bias. . . . . . . . . . . 703.6 Nonlinear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.7 Lagged Dependent Variables. . . . . . . . . . . . . . . . . . . . . . 793.8 Dummy Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.9 Regression with Interaction Variables . . . . . . . . . . . . . . . 923.10 Regression Approach to Investigating the Effect

of Alternative Business Strategies . . . . . . . . . . . . . . . . . . 963.11 Logistic Regression and Credit Risk Analysis:

Ohlson’s and Shumway’s Methods for EstimatingDefault Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Appendix 1: Dynamic Ratio Analysis . . . . . . . . . . . . . . . . . . . . . 100Appendix 2: Term Structure of Interest Rate. . . . . . . . . . . . . . . . 100Appendix 3: Partial Adjustment Dividend BehaviorModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Appendix 4: Logistic Model and Probit Model . . . . . . . . . . . . . . 108

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Appendix 5: SAS Code for Hazard Model inBankruptcy Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4 Simultaneous Equation Models. . . . . . . . . . . . . . . . . . . . . . . . . 1154.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.2 Discussion of Simultaneous Equation System . . . . . . . . . 1164.3 Two-Stage and Three-Stage Least Squares Method. . . . . 116

4.3.1 Identification Problem . . . . . . . . . . . . . . . . . . . 1174.3.2 Two-Stage Least Squares . . . . . . . . . . . . . . . . . 1194.3.3 Three-Stage Least Squares . . . . . . . . . . . . . . . . 119

4.4 Application of Simultaneous Equation in FinanceResearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5 Econometric Approach to Financial Analysis, Planning,and Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.2 Simultaneous Nature of Financial Analysis, Planning,

and Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.2.1 Basic Concepts of Simultaneous

Econometric Models . . . . . . . . . . . . . . . . . . . . 1265.2.2 Interrelationship of Accounting

Information . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.2.3 Interrelationship of Financial Policies . . . . . . . 127

5.3 The Simultaneity and Dynamicsof Corporate-Budgeting Decisions. . . . . . . . . . . . . . . . . . 1275.3.1 Definitions of Endogenous and Exogenous

Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.3.2 Model Specification and Applications . . . . . . . 127

5.4 Applications of SUR Estimation Method in FinancialAnalysis and Planning. . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.4.1 The Role of Firm-Related Variables

in Capital Asset Pricing . . . . . . . . . . . . . . . . . . 1375.4.2 The Role of Capital Structure

in Corporate-Financing Decisions . . . . . . . . . . 1405.5 Applications of Structural Econometric Models

in Financial Analysis and Planning . . . . . . . . . . . . . . . . . 1415.5.1 A Brief Review . . . . . . . . . . . . . . . . . . . . . . . . 1415.5.2 AT&T’s Econometric Planning Model. . . . . . . 142

5.6 Programming Versus Simultaneous VersusEconometric Financial Models . . . . . . . . . . . . . . . . . . . . 142

5.7 Financial Analysis and Business Policy Decisions . . . . . 1445.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145Appendix: Johnson & Johnson as a Case Study . . . . . . . . . . . . . 146Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

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6 Fixed Effects Versus Random Effectsin Finance Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1596.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1606.2 The Dummy Variable Technique and the Error

Component Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1606.3 Impacts of Firm Effect and Time Effect on Stock Price

Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1626.4 Functional Form and Pooled Time-Series

and Cross-Sectional Data . . . . . . . . . . . . . . . . . . . . . . . . 1646.5 Clustering Effect and Clustered Standard Errors . . . . . . . 1706.6 Hausman Test for Determining Either Fixed Effects

Model or Random Effects Model . . . . . . . . . . . . . . . . . . 1706.7 Efficient Firm Fixed Effects Estimator and Efficient

Correlated Random Effects Estimator . . . . . . . . . . . . . . . 1716.8 Empirical Evidence of Optimal Payout Ratio Under

Uncertainty and the Flexibility Hypothesis . . . . . . . . . . . 1716.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175Appendix: Optimal Payout Ratio Under Uncertainty andthe Flexibility Hypothesis: Theory and Empirical Evidence . . . . 175Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7 Alternative Methods to Deal with Measurement Error. . . . . . 1817.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1827.2 Effects of Errors-in-Variables in Different Cases . . . . . . . 183

7.2.1 Bivariate Normal Case . . . . . . . . . . . . . . . . . . . 1837.2.2 Multivariate Case . . . . . . . . . . . . . . . . . . . . . . . 183

7.3 Estimation Methods When Variables Are Subjectto Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1857.3.1 Classical Estimation Method . . . . . . . . . . . . . . 1857.3.2 Grouping Method. . . . . . . . . . . . . . . . . . . . . . . 1887.3.3 Instrumental Variable Method . . . . . . . . . . . . . 1897.3.4 Mathematical Method . . . . . . . . . . . . . . . . . . . 1907.3.5 Maximum Likelihood Method . . . . . . . . . . . . . 1927.3.6 LISREL and MIMIC Methods . . . . . . . . . . . . . 1937.3.7 Bayesian Approach . . . . . . . . . . . . . . . . . . . . . 194

7.4 Applications of Errors-in-Variables Models in FinanceResearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1957.4.1 Cost of Capital. . . . . . . . . . . . . . . . . . . . . . . . . 1957.4.2 Capital Asset Pricing Model . . . . . . . . . . . . . . 1997.4.3 Capital Structure . . . . . . . . . . . . . . . . . . . . . . . 2047.4.4 Measurement Error in Investment

Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2057.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

8 Three Alternative Methods in Testing Capital Asset PricingModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2118.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

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8.2 Empirical Test on Capital Asset Pricing Model. . . . . . . . 2138.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2138.2.2 Grouping Method for Testing Capital Asset

Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . 2148.2.3 Instrumental Variable Method for Testing

Capital Asset Pricing Model . . . . . . . . . . . . . . 2188.2.4 Applying Instrumental Variable Methods

into Grouping Sample . . . . . . . . . . . . . . . . . . . 2218.2.5 Maximum Likelihood Method for Testing

Capital Asset Pricing Model . . . . . . . . . . . . . . 2218.2.6 Asset Pricing Model Tests with Individual

Stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2248.3 Normality Test for Time-Series Estimators and Future

Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2248.4 The Investment Horizon of Beta Estimation . . . . . . . . . . 2268.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

9 Spurious Regression and Data Mining in Conditional AssetPricing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2439.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2449.2 Model Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2449.3 Spurious Regression and Data Mining in Predictive

Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2469.4 Spurious Regression, Data Mining, and Conditional

Asset Pricing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2469.5 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2489.6 The Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

9.6.1 Predictive Regressions . . . . . . . . . . . . . . . . . . . 2509.6.2 Conditional Asset Pricing Models . . . . . . . . . . 251

9.7 Results for Predictive Regressions. . . . . . . . . . . . . . . . . . 2529.7.1 Pure Spurious Regression . . . . . . . . . . . . . . . . 2529.7.2 Spurious Regression and Data Mining . . . . . . . 256

9.8 Results for Conditional Asset Pricing Models . . . . . . . . . 2619.8.1 Cases with Small Amounts of Persistence . . . . 2619.8.2 Cases with Persistence . . . . . . . . . . . . . . . . . . . 2619.8.3 Suppressing Time-Varying Alphas. . . . . . . . . . 2649.8.4 Suppressing Time-Varying Betas . . . . . . . . . . . 2659.8.5 A Cross Section of Asset Returns . . . . . . . . . . 2679.8.6 Revisiting Previous Evidence. . . . . . . . . . . . . . 267

9.9 Solutions to the Problems of Spurious Regressionand Data Mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2699.9.1 Solutions in Predictive Regressions . . . . . . . . . 2699.9.2 Solutions in Conditional Asset Pricing

Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2719.10 Robustness of the Asset Pricing Results . . . . . . . . . . . . . 271

9.10.1 Multiple Instruments . . . . . . . . . . . . . . . . . . . . 271

Contents xiii

9.10.2 Multiple-Beta Models . . . . . . . . . . . . . . . . . . . 2719.10.3 Predicting the Market Return . . . . . . . . . . . . . . 2729.10.4 Simulations Under the Alternative

Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2729.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

Part II Time-Series Analysis and Its Applications

10 Time Series: Analysis, Model, and Forecasting . . . . . . . . . . . . 27910.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28010.2 The Classical Time-Series Component Model . . . . . . . . . 28010.3 Moving Average and Seasonally Adjusted

Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28510.4 Linear and Log Linear Time Trend Regressions . . . . . . . 28810.5 Exponential Smoothing and Forecasting . . . . . . . . . . . . . 29410.6 Autoregressive Forecasting Model . . . . . . . . . . . . . . . . . 30010.7 ARIMA Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30310.8 Autoregressive Conditional Heteroscedasticity . . . . . . . . 306

10.8.1 Autoregressive ConditionalHeteroscedasticity (ARCH) Models . . . . . . . . . 306

10.8.2 Generalized Autoregressive ConditionalHeteroscedasticity (GARCH) Model . . . . . . . . 306

10.8.3 The GARCH Universe. . . . . . . . . . . . . . . . . . . 30610.9 Composite Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . 308

10.9.1 Composite Forecasting of Livestock Prices . . . 30810.9.2 Combined Forecasting of the Taiwan

Weighted Stock Index . . . . . . . . . . . . . . . . . . . 30910.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309Appendix 1: The Holt–Winters Forecasting Model forSeasonal Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310Appendix 2: Composite Forecasting Method. . . . . . . . . . . . . . . . 314Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

11 Hedge Ratio and Time-Series Analysis . . . . . . . . . . . . . . . . . . 31711.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31811.2 Alternative Theories for Deriving the Optimal Hedge

Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32011.2.1 Static Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32011.2.2 Dynamic Case . . . . . . . . . . . . . . . . . . . . . . . . . 32411.2.3 Case with Production and Alternative

Investment Opportunities . . . . . . . . . . . . . . . . . 32511.3 Alternative Methods for Estimating the Optimal Hedge

Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32611.3.1 Estimation of the Minimum-Variance (MV)

Hedge Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 32611.3.2 Estimation of the Optimum Mean-Variance

and Sharpe Hedge Ratios . . . . . . . . . . . . . . . . . 329

xiv Contents

11.3.3 Estimation of the Maximum Expected UtilityHedge Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

11.3.4 Estimation of Mean Extended-Gini (MEG)Coefficient-Based Hedge Ratios . . . . . . . . . . . . 330

11.3.5 Estimation of Generalized Semivariance(GSV) Based Hedge Ratios . . . . . . . . . . . . . . . 331

11.4 Hedging Horizon, Maturity of Futures Contract, DataFrequency, and Hedging Effectiveness . . . . . . . . . . . . . . 331

11.5 Empirical Results of Hedge Ratio Estimation . . . . . . . . . 33211.5.1 OLS Method . . . . . . . . . . . . . . . . . . . . . . . . . . 33311.5.2 ARCH GARCH . . . . . . . . . . . . . . . . . . . . . . . . 33311.5.3 EGARCH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33311.5.4 GJR-GARCH. . . . . . . . . . . . . . . . . . . . . . . . . . 33411.5.5 TGARCH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

11.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335Appendix 1: Theoretical Models . . . . . . . . . . . . . . . . . . . . . . . . . 337Appendix 2: Empirical Models . . . . . . . . . . . . . . . . . . . . . . . . . . 339Appendix 3: Monthly Data of S&P 500 Indexand Its Futures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

Part III Statistical Distributions, Option Pricing Modeland Risk Management

12 The Binomial, Multinomial Distributions, and OptionPricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35712.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35712.2 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 35812.3 The Simple Binomial Option Pricing Model . . . . . . . . . . 36112.4 The Generalized Binomial Option Pricing Model . . . . . . 36412.5 Multinomial Option Pricing Model . . . . . . . . . . . . . . . . . 368

12.5.1 Derivation of the Option Pricing Model. . . . . . 36812.5.2 The Black and Scholes Model as a Limiting

Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36912.6 A Lattice Framework for Option Pricing. . . . . . . . . . . . . 371

12.6.1 Modification of the Two-State Approachfor a Single-State Variable . . . . . . . . . . . . . . . . 371

12.6.2 A Lattice Model for Valuation of Optionson Two Underlying Assets. . . . . . . . . . . . . . . . 373

12.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

13 Two Alternative Binomial Option Pricing ModelApproaches to Derive Black–Scholes Option PricingModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37913.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37913.2 The Two-State Option Pricing Model of Rendleman

and Bartter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

Contents xv

13.2.1 The Discrete-Time Model . . . . . . . . . . . . . . . . 38013.2.2 The Continuous Time Model . . . . . . . . . . . . . . 382

13.3 The Binomial Option Pricing Model of CRR . . . . . . . . . 38513.3.1 The Binomial Option Pricing Formula

of CRR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38513.3.2 Limiting Case . . . . . . . . . . . . . . . . . . . . . . . . . 385

13.4 Comparison of the Two Approaches . . . . . . . . . . . . . . . . 38813.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389Appendix: The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . 389Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

14 Normal, Lognormal Distribution, and Option PricingModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39314.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39414.2 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 39414.3 The Lognormal Distribution . . . . . . . . . . . . . . . . . . . . . . 39514.4 The Lognormal Distribution and Its Relationship

to the Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . 39614.5 Multivariate Normal and Lognormal Distributions . . . . . 39714.6 The Normal Distribution as an Application

to the Binomial and Poisson Distributions . . . . . . . . . . . 39914.7 Applications of the Lognormal Distribution in Option

Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40214.8 The Bivariate Normal Density Function . . . . . . . . . . . . . 40314.9 American Call Options . . . . . . . . . . . . . . . . . . . . . . . . . . 405

14.9.1 Price American Call Options by the BivariateNormal Distribution . . . . . . . . . . . . . . . . . . . . . 405

14.9.2 Pricing an American Call Option:An Example. . . . . . . . . . . . . . . . . . . . . . . . . . . 406

14.10 Price Bounds for Options . . . . . . . . . . . . . . . . . . . . . . . . 40914.10.1 Options Written on Nondividend-Paying

Stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40914.10.2 Options Written on Dividend-Paying

Stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41014.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413Appendix 1: Microsoft Excel Program for CalculatingCumulative Bivariate Normal Density Function . . . . . . . . . . . . . 414Appendix 2: Microsoft Excel Program for Calculating theAmerican Call Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

15 Copula, Correlated Defaults, and Credit VaR. . . . . . . . . . . . . 41915.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42015.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

15.2.1 CreditMetrics . . . . . . . . . . . . . . . . . . . . . . . . . . 42115.2.2 Copula Function . . . . . . . . . . . . . . . . . . . . . . . 42415.2.3 Factor Copula Model . . . . . . . . . . . . . . . . . . . . 426

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15.3 Experimental Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42715.3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42715.3.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42915.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430

15.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

16 Multivariate Analysis: Discriminant Analysis and FactorAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43916.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43916.2 Important Concepts of Linear Algebra . . . . . . . . . . . . . . 44016.3 Two-Group Discriminant Analysis . . . . . . . . . . . . . . . . . 44516.4 k-Group Discriminant Analysis . . . . . . . . . . . . . . . . . . . . 44916.5 Factor Analysis and Principal Component Analysis . . . . 45116.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451Appendix 1: Relationship Between Discriminant Analysisand Dummy Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . 452Appendix 2: Principal Component Analysis . . . . . . . . . . . . . . . . 454Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

Part IV Statistics, Itô’s Calculus and Option Pricing Model

17 Stochastic Volatility Option Pricing Models . . . . . . . . . . . . . . 46117.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46117.2 Nonclosed-Form Type of Option Pricing Model . . . . . . . 46217.3 Review of Characteristic Function. . . . . . . . . . . . . . . . . . 46617.4 Closed-Form Type of Option Pricing Model. . . . . . . . . . 46717.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471Appendix: The Market Price of the Risk. . . . . . . . . . . . . . . . . . . 471Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

18 Alternative Methods to Estimate Implied Variance: Reviewand Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47318.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47318.2 Numerical Search Method and Closed-Form

Derivation Method to Estimate Implied Variance . . . . . . 47418.3 MATLAB Approach to Estimate Implied Variance. . . . . 48118.4 Approximation Approach to Estimate Implied

Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48318.5 Some Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . 487

18.5.1 Cases from USA—IndividualStock Options . . . . . . . . . . . . . . . . . . . . . . . . . 487

18.5.2 Cases from China—ETF 50 Options . . . . . . . . 48718.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

Contents xvii

19 Numerical Valuation of Asian Options with HigherMoments in the Underlying Distribution . . . . . . . . . . . . . . . . . 49119.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49219.2 Definitions and the Basic Binomial Model . . . . . . . . . . . 49319.3 Edgeworth Binomial Model for Asian Option

Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49419.4 Upper Bound and Lower Bound for European-Asian

Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49719.5 Upper Bound and Lower Bound for American-Asian

Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50019.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

19.6.1 Pricing European-Asian OptionsUnder Lognormal Distribution . . . . . . . . . . . . . 502

19.6.2 Pricing American-Asian Options UnderLognormal Distribution . . . . . . . . . . . . . . . . . . 506

19.6.3 Pricing European-Asian OptionsUnder Distributions with Higher Moments . . . 510

19.6.4 Pricing American-Asian Options UnderDistributions with Higher Moments . . . . . . . . . 513

19.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

20 Itô’s Calculus: Derivation of the Black–Scholes OptionPricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51720.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51820.2 The Itô Process and Financial Modeling . . . . . . . . . . . . . 51820.3 Itô Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52120.4 Stochastic Differential Equation Approach

to Stock-Price Behavior . . . . . . . . . . . . . . . . . . . . . . . . . 52220.5 The Pricing of an Option . . . . . . . . . . . . . . . . . . . . . . . . 52620.6 A Reexamination of Option Pricing . . . . . . . . . . . . . . . . 52920.7 Remarks on Option Pricing . . . . . . . . . . . . . . . . . . . . . . . 53220.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534Appendix: An Alternative Method to Derive theBlack–Scholes Option Pricing Model . . . . . . . . . . . . . . . . . . . . . 534Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

21 Alternative Methods to Derive Option Pricing Models . . . . . . 54121.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54221.2 A Brief Review of Alternative Approaches

for Deriving Option Pricing Model . . . . . . . . . . . . . . . . . 54421.2.1 Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . 54421.2.2 Black–Scholes Model. . . . . . . . . . . . . . . . . . . . 547

21.3 Relationship Between Binomial OPMand Black–Scholes OPM. . . . . . . . . . . . . . . . . . . . . . . . . 547

xviii Contents

21.4 Compare Cox et al. and Rendleman and BartterMethods to Derive OPM. . . . . . . . . . . . . . . . . . . . . . . . . 55121.4.1 Cox et al. Method . . . . . . . . . . . . . . . . . . . . . . 55121.4.2 Rendleman and Bartter Method . . . . . . . . . . . . 554

21.5 Lognormal Distribution Approach to DeriveBlack–Scholes Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 558

21.6 Using Stochastic Calculus to Derive Black–ScholesModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

21.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564Appendix: The Relationship Between Binomial Distributionand Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

22 Constant Elasticity of Variance Option Pricing Model:Integration and Detailed Derivation . . . . . . . . . . . . . . . . . . . . . 57122.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57122.2 The CEV Diffusion and Its Transition Probability

Density Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57222.3 Review of Noncentral Chi-Square Distribution . . . . . . . . 57422.4 The Noncentral Chi-Square Approach to Option

Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57522.4.1 Detailed Derivations of C1 and C2 . . . . . . . . . . 57522.4.2 Some Computational Considerations . . . . . . . . 579

22.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580Appendix: Proof of Feller’s Lemma . . . . . . . . . . . . . . . . . . . . . . 580Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582

23 Option Pricing and Hedging Performance Under StochasticVolatility and Stochastic Interest Rates . . . . . . . . . . . . . . . . . . 58323.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58423.2 The Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . 587

23.2.1 Pricing Formula for European Options. . . . . . . 58823.2.2 Hedging and Hedge Ratios . . . . . . . . . . . . . . . 59023.2.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . 594

23.3 Data Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59523.4 Empirical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597

23.4.1 Static Performance . . . . . . . . . . . . . . . . . . . . . . 59823.4.2 Dynamic Hedging Performance . . . . . . . . . . . . 60323.4.3 Regression Analysis of Option Pricing and

Hedging Errors . . . . . . . . . . . . . . . . . . . . . . . . 61223.4.4 Robustness of Empirical Results . . . . . . . . . . . 614

23.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617Appendix 1: Derivation of Stochastic Interest Modeland Stochastic Volatility Model. . . . . . . . . . . . . . . . . . . . . . . . . . 617Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619

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24 Nonparametric Method for European Option Bounds . . . . . . 62324.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62324.2 The Bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62424.3 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62824.4 Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63224.5 Empirical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63424.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639Appendix 1: Related Option Studies AdoptingNonparametric Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640Appendix 2: Asset Pricing Model with a StochasticKernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653

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