Notes econometricswithr

Econometric Analysis of Financial Market Data1

ZONGWU CAI

E-mail address: zcai@uncc.edu
Department of Mathematics & Statistics
University of North Carolina, Charlotte, NC 28223, U.S.A.

September 19, 2012

c 2012, ALL RIGHTS RESERVED by ZONGWU CAI

1 This manuscript may be printed and reproduced for individual or instructional use, but may
not be printed for commercial purposes.

i

Preface
The main purpose of this lecture notes is to provide you with a foundation to pursue the
basic theory and methodology as well as applied projects involving the skills to analyzing
financial data. This course also gives an overview of the econometric methods (models and
their modeling techniques) applicable to financial economic modeling. More importantly, it
is the ultimate goal of bringing you to the research frontier of the empirical (quantitative)
finance. To model financial data, some packages will be used such as R, which is a very
convenient programming language for doing homework assignments and projects. You can
download it for free from the web site at http://www.r-project.org/.

Several projects, including the heavy computer works, are assigned throughout the semester.
The group discussion is allowed to do the projects and the computer related homework, par-
ticularly writing the computer codes. But, writing the final report to each project or home
assignment must be in your own language. Copying each other will be regarded as a cheating.
If you use the R language, similar to SPLUS, you can download it from the public web site
at http://www.r-project.org/ and install it into your own computer or you can use PCs at
our labs. You are STRONGLY encouraged to use (but not limited to) the package R since
it is a very convenient programming language for doing statistical analysis and Monte Carol
simulations as well as various applications in quantitative economics and finance. Of course,
you are welcome to use any one of other packages such as SAS, MATLAB, GAUSS, and
STATA. But, I might not have an ability of giving you a help if doing so.

How to Install R ?
The main package used is R, which is free from R-Project for Statistical Computing.

(1) go to the web site http://www.r-project.org/;

(2) click CRAN;

(3) choose a site for downloading, say http://cran.cnr.Berkeley.edu;

(4) click Windows (95 and later);

(5) click base;

(6) click R-2.15.1-win.exe (Version of 06-22-2012) to save this file first and then run it
to install (Note that it is updated almost every three months).

The above steps install the basic R into your computer. If you need to install other
packages, you need to do the followings:

(7) After it is installed, there is an icon on the screen. Click the icon to get into R;

(8) Go to the top and find packages and then click it;

(9) Go down to Install package(s)... and click it;

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(10) There is a new window. Choose a location to download packages, say USA(CA1),
move mouse to there and click OK;

(11) There is a new window listing all packages. You can select any one of packages and
click OK, or you can select all of them and then click OK.

Data Analysis and Graphics Using R – An Introduction (109 pages)

I encourage you to download the ﬁle r-notes.pdf (109 pages) which can be downloaded
from http://www.math.uncc.edu/˜ zcai/r-notes.pdf and learn it by yourself. Please
see me if any questions.

CRAN Task View: Empirical Finance
This CRAN Task View contains a list of packages useful for empirical work in Finance and
it can be downloaded from the web site at
http://cran.cnr.berkeley.edu/src/contrib/Views/Finance.html.

CRAN Task View: Computational Econometrics
Base R ships with a lot of functionality useful for computational econometrics, in particular
in the stats package. This functionality is complemented by many packages on CRAN. It
can be downloaded from the web site at
http://cran.cnr.berkeley.edu/src/contrib/Views/Econometrics.html.

Contents

1 A Motivation Example 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Preliminary Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Jump-Diffusion Modeling Procedures . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Pricing American-style Options Using Stratification Simulation Method . . . 8
1.5 Hedging Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Basic Concepts of Prices and Returns 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Time Value of Money . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 Assets and Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.3 Financial Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Statistical Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.2 Frequency of Observations . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.3 Definition of Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Stylized Facts for Financial Returns . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Linear Time Series Models and Their Applications 31
3.1 Stationary Stochastic Process . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Constant Expected Return Model . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.2 Regression Model Representation . . . . . . . . . . . . . . . . . . . . 34
3.2.3 CER Model of Asset Returns and Random Walk Model of Asset Prices 35
3.2.4 Monte Carlo Simulation Method . . . . . . . . . . . . . . . . . . . . . 36
3.2.5 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.6 Statistical Properties of Estimates . . . . . . . . . . . . . . . . . . . . 38
3.3 AR(1) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.1 Estimation and Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.2 White Noise Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 41

iii

CONTENTS iv

3.3.3 Unit Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.4 Estimation and Tests in the Presence of a Unit Root . . . . . . . . . 42
3.4 MA(1) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 ARMA, ARIMA, and ARFIMA Processes . . . . . . . . . . . . . . . . . . . 45
3.5.1 ARMA(1,1) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5.2 ARMA(p,q) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.5.3 AR(p) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.5.4 MA(q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5.5 AR(∞) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5.6 MA(∞) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5.7 ARIMA Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5.8 ARFIMA Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.6 R Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.7 Regression Models With Correlated Errors . . . . . . . . . . . . . . . . . . . 56
3.8 Seasonal Time Series Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.9 Consistent Estimation of Covariance Matrix . . . . . . . . . . . . . . . . . . 56
3.10 Comments on Nonlinear Models and Their Applications . . . . . . . . . . . . 57
3.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.11.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.11.2 R Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.12 Appendix A: Linear Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.13 Appendix B: Forecasting Based on AR(p) Model . . . . . . . . . . . . . . . . 62
3.14 Appendix C: Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.15 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Predictability of Asset Returns 69
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.1 Martingale Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.2 Tests of MD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Random Walk Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2.1 IID Increments (RW1) . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2.2 Independent Increments (RW2) . . . . . . . . . . . . . . . . . . . . . 71
4.2.3 Uncorrelated Increments (RW3) . . . . . . . . . . . . . . . . . . . . . 72
4.2.4 Unconditional Mean is the Best Predictor (RW4) . . . . . . . . . . . 72
4.3 Tests of Predictability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.1 Nonparametric Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3.2 Autocorrelation Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.3 Variance Ratio Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3.4 Trading Rules and Market Eﬃciency . . . . . . . . . . . . . . . . . . 80
4.4 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4.1 Evidence About Returns Predictability Using VR and Autocorrelation Tests 84
4.4.2 Cross Lag Autocorrelations and Lead-Lag Relations . . . . . . . . . . 85
4.4.3 Evidence About Returns Predictability Using Trading Rules . . . . . 86
4.5 Predictability of Real Stock and Bond Returns . . . . . . . . . . . . . . . . . 87
4.5.1 Financial Predictors . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

CONTENTS v

4.5.2 Models and Modeling Methods . . . . . . . . . . . . . . . . . . . . . 88
4.6 A Recent Perspective on Predictability of Asset Return . . . . . . . . . . . . 95
4.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.6.2 Conditional Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.6.3 Conditional Variances . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.6.4 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.6.5 The future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.7 Comments on Predictability Based on Nonlinear Models . . . . . . . . . . . 101
4.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.8.1 Exercises for Homework . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.8.2 R Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.8.3 Project #1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5 Market Model 111
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.2 Assumptions About Asset Returns . . . . . . . . . . . . . . . . . . . . . . . 112
5.3 Unconditional Properties of Returns . . . . . . . . . . . . . . . . . . . . . . . 112
5.4 Conditional Properties of Returns . . . . . . . . . . . . . . . . . . . . . . . . 113
5.5 Beta as a Measure of Portfolio Risk . . . . . . . . . . . . . . . . . . . . . . . 114
5.6 Diagnostics for Constant Parameters . . . . . . . . . . . . . . . . . . . . . . 115
5.7 Estimation and Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . 116
5.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6 Event-Study Analysis 119
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2 Outline of an Event Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.3 Models for Measuring Normal Returns . . . . . . . . . . . . . . . . . . . . . 121
6.4 Measuring and Analyzing Abnormal Returns . . . . . . . . . . . . . . . . . . 122
6.4.1 Estimation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.4.2 Aggregation of Abnormal Returns . . . . . . . . . . . . . . . . . . . . 124
6.4.3 Modifying the Null Hypothesis: . . . . . . . . . . . . . . . . . . . . . 127
6.4.4 Nonparametric Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.4.5 Cross-Sectional Models . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.4.6 Power of Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.5 Further Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7 Introduction to Portfolio Theory 136
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.1.1 Efficient Portfolios With Two Risky Assets . . . . . . . . . . . . . . . 137
7.1.2 Efficient Portfolios with One Risky Asset and One Risk-Free Asset . . 138
7.1.3 Efficient portfolios with two risky assets and a risk-free asset . . . . . 139

CONTENTS vi

7.2 Eﬃcient Portfolios with N risky assets . . . . . . . . . . . . . . . . . . . . . 140
7.3 Another Look at Mean-Variance Eﬃciency . . . . . . . . . . . . . . . . . . . 142
7.4 The Black-Litterman Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.4.1 Expected Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.4.2 The Black-Litterman Model . . . . . . . . . . . . . . . . . . . . . . . 145
7.4.3 Building the Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.5 Estimation of Covariance Matrix . . . . . . . . . . . . . . . . . . . . . . . . 147
7.5.1 Estimation Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.5.2 Shrinkage estimator of the covariance matrix . . . . . . . . . . . . . . 150
7.5.3 Recent Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

8 Capital Asset Pricing Model 155
8.1 Review of the CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.2 Statistical Framework for Estimation and Testing . . . . . . . . . . . . . . . 157
8.2.1 Time-Series Regression . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8.2.2 Cross-Sectional Regression . . . . . . . . . . . . . . . . . . . . . . . . 159
8.2.3 Fama-MacBeth Procedure . . . . . . . . . . . . . . . . . . . . . . . . 162
8.3 Empirical Results on CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8.3.1 Testing CAPM Based On Cross-Sectional Regressions . . . . . . . . . 163
8.3.2 Return-Measurement Interval and Beta . . . . . . . . . . . . . . . . . 165
8.3.3 Results of FF and KSS . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
8.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

9 Multifactor Pricing Models 169
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
9.1.1 Why Do We Expect Multiple Factors? . . . . . . . . . . . . . . . . . 169
9.1.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
9.2 Selection of Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
9.2.1 Theoretical Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 171
9.2.2 Small and Value/Growth Stocks . . . . . . . . . . . . . . . . . . . . . 171
9.2.3 Macroeconomic Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 172
9.2.4 Statistical Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 176
9.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
9.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

List of Tables

2.1 Illustration of the Eﬀects of Compounding: . . . . . . . . . . . . . . . . . . . 15

3.1 Deﬁnitions of ten types of stochastic process . . . . . . . . . . . . . . . . . . 32
3.2 Large-sample critical values for the ADF statistic . . . . . . . . . . . . . . . 43
3.3 Summary of DF test for unit roots in the absence of serial correlation . . . . 44

4.1 Variance ratio test values, daily 1991-2000 (from Taylor, 2005) . . . . . . . . 86
4.2 Variance ratio test values, weekly 1962-1994 (from Taylor, 2005) . . . . . . . 86
4.3 Autocorrelations in daily, weekly, and monthly stock index returns . . . . . . 87

7.1 Example Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.2 Expected excess return vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.3 Recommended portfolio weights . . . . . . . . . . . . . . . . . . . . . . . . . 146

vii

List of Figures

1.1 The time series plot of the swap rates. . . . . . . . . . . . . . . . . . . . . . 3
1.2 The time series plot of the log of swap rates. . . . . . . . . . . . . . . . . . . 4
1.3 The scatter plot of the log return versus the level of log of swap rates. . . . . 5

2.1 The weekly and monthly prices of IBM stock. . . . . . . . . . . . . . . . . . 18
2.2 The weekly and monthly returns of IBM stock. . . . . . . . . . . . . . . . . . 20
2.3 The empirical distribution of standardized IBM daily returns and the pdf of standard normal. N
2.4 The empirical distribution of standardized Microsoft daily returns and the pdf of standard norma
2.5 Q-Q plots for the standardized IBM returns (top panel) and the standardized Microsoft returns (

3.1 Some examples of diﬀerent categories of stochastic processes. . . . . . . . . . 33
3.2 Relationships between categories of uncorrelated processes. . . . . . . . . . . 33
3.3 Monte Carlo Simulation of the CER model. . . . . . . . . . . . . . . . . . . 37
3.4 Sample autocorrelation function of the absolute series of daily simple returns for the CRSP value

6.1 Time Line of an event study. . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2 Power function of the J1 test at the 5% signiﬁcance level for sample sizes 1, 10, 20 and 50.133

7.1 Plot of portfolio expected return, µp versus portfolio standard deviation, σp . . 137
7.2 Plot of portfolio expected return versus standard deviation. . . . . . . . . . . 139
7.3 Plot of portfolio expected return versus standard deviation. . . . . . . . . . . 140
7.4 Deriving the new combined return vector E(R). . . . . . . . . . . . . . . . . 148

8.1 Cross-sectional regression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

viii

Chapter 1

A Motivation Example

The purpose of this chapter is to present you, as a motivation example, a simple procedure
that can be used for proposing a reasonable jump-diffusion model for a real market data
(swap rates), calibrating parameters for the jump-diffusion model, and pricing American-
style options under the proposed jump-diffusion process. In addition, we will discuss hedging
issues for such options and sensitivity of parameters for American-style options.

1.1 Introduction
It is well known (see, e.g., Duffie (1996)) that under some regular conditions, there is
an equivalent martingale measure Q, such that for any European contingent claim on an
underlying {Xt ; t ≥ 0} without paying dividends with maturity T in the market, it can be
priced as follows:
T
P (0, T ) = EQ
0 exp − r(s)ds g(XT , T ) , (1.1)
0
where g(·, ·) stands for the payoff function of underling for this contingent claim, P (0, T ) is
the claim’s arbitrage-free or fair price at time 0, rt is the riskless short-term interest rate,
and EQ [·] presents the expectation operator conditional on the information up to now.
0

For an American contingent claim on the same underlying with maturity T in the market,
it can be priced similarly:
τ
P (0, T ) = sup EQ exp −
0 r(s)ds g(Xτ , τ ) , (1.2)
τ ∈Γ 0

where Γ is the collection of all stopping times less than the maturity time T . A comparison
of (1.1) with (1.2) reveals that the theory is similar but the computing for the American
option is much difficult.

1

CHAPTER 1. A MOTIVATION EXAMPLE 2

This theory provides us a “risk neutral” scheme to price any contingent claim. More
precisely, we can pretend to live a risk neutral world to modelling and calibrating parameters
using the data “lived” in the real world, then we do pricing using equation (1.1) or (1.2).
We will use this scheme throughout this paper.

In this chapter, we will present a simple procedure that can be used for proposing a
reasonable jump-diffusion model for a real market data, calibrating parameters for the jump-
diffusion model, and pricing American-style options under the proposed jump-diffusion pro-
cess. In addition, we will discuss hedging issues for such options and sensitivity of parameters
for American-style options. The remainder of the chapter is structured as follows. Section
2 presents some empirical properties of the data by graphing, mining data, and doing some
preliminary statistical analysis. Section 3 provides a jump-diffusion model based on the giv-
en properties observed from Section 2, a calibration of parameters under this jump-diffusion
setting by MLE method, and a test for the existence of “jump”. Section 4 proposes a uni-
versal algorithm for American-style options with one-factor underlying model, and uses this
algorithm to price American option for the real data. Section 5 presents hedging issues for
the given American option. Section 6 concludes this chapter and discusses an extension of
our model to a more general tractable jump-diffusion setting called “affine jump-diffusion”
model proposed by Duffie, Pan and Singleton (2000).

1.2 Preliminary Statistical Analysis
The data we will investigate is a collection of swap rates (the differences between 10 years
LIBOR rates and 10-year treasury bond’s yields) from December 19, 2002 to October 15,
2004. We can present the data graphically in Figure 1.1 by the time series plot. From the
graph, we observe the followings:

(O1) We can visually find there are some possible jumps for swap rates, and jumps seem
to have almost same frequencies for positive and negative jumps. In addition, from
economic standpoint of view, the difference of LIBOR and treasury yield should always
be positive since the former always includes some credit issues.

(O2) We can find “mean reversion” from the graph, which means a very high swap rate tends
to go lower, while a low swap rate tends to bounce back to a higher level. Economically,


65
60
55
Swap rate

50
45
40
35
30

0 100 200 300 400

date

Figure 1.1: The time series plot of the swap rates.

it makes sense since we can not expect a sequence of swap rates going up without any
pull-back.

(O3) We shall pay attention to the graph not exactly presenting the data, since we don’t
consider irregularity of time space at the x-axis because of no recording of holidays and
weekends. For details on the calendar effects, see the book by Taylor (2005, Section
4.5). The implication of this irregularity of time space is that some of possible jumps
maybe come from no transaction for long-time which leads to an accumulative effects
of a series bad or good news on the next transaction day(s).

(O4) We can find visually that jumps seem to be clustered, which means that if a jump
occurs, there will follow more jumps with a greater probability, and a sequential positive
jumps occurred will follow a a sequential negative jumps with a greater probability.
This is an embarrassing finding, since we will not deal with this issue in this chapter
but this is an important research topic for academics and practitioners.


A formal way to modelling a dynamic system for a necessary positive data is to modelling
the logarithm of the original data. The transformed data is graphed in Figure 1.2: Since our

4.2
4.0
Log of Swap Rate

3.8
3.6

0 100 200 300 400

date

Figure 1.2: The time series plot of the log of swap rates.

objective is to modelling a dynamic mechanism of the evolution of swap rates, we propose a
general stochastic differential equation to the transformed variable (logarithm of swap rate)
which can usually be called “state” variable. Let St be the swap rate at time t, and denote
Xt as the logarithm of St , namely, Xt = log(St ). The general stochastic differential equation
(SDE, or called Black-Scholes model) of Xt is as follows,

dXt = µ(Xt )dt + σ(Xt )dWt + dJt , X0 = x0 , (1.3)

where µ(·) (drift) and σ(·) (diffusion) stand for instantaneous mean function and volatility
function of the process respectively and Wt and Jt are a standard Brownian motion and a
pure jump process respectively.

The objective of modelling, in fact, is to specify the explicit forms of µ(·) and σ(·), and
probability mechanism of pure jump process Jt . In this section, we will have some idea about
the possible shape of σ(·) by a preliminary approximation of the SDE and the transformed


data. First, for a very small time interval δt, the SDE can be approximated by a difference
equation (Euler approximation) as follows,

Xt+δt − Xt ≃ µ(Xt )δt + σ(Xt )(Wt+δt − Wt ) + (Jt+δt − Jt )
≃ σ(Xt )(Wt+δt − Wt ) + (Jt+δt − Jt ). (1.4)

The reason to omit the term µ(Xt )δt in the above equation is that this term is of the order
of o(1) while other two terms have a lower order. By (1.4), we can have a preliminary
visual sense of the form of σ(·) by looking at the graph of the transformed data with Xt as
x-coordinate and Xt+1 − Xt (log return) as y-coordinate; see Figure 1.3. The theory behind
0.1
0.0
Log return

−0.1
−0.2

3.6 3.8 4.0 4.2

Level of log swap rate

Figure 1.3: The scatter plot of the log return versus the level of log of swap rates.

this idea can be found in Stanton (1997) or Cai and Hong (2003). We will discuss this idea
in detail later. In the above figure, each horizontal line except x-axis represents the level of
number of standard deviations away from zero. Except some outliers which can be explained
partly by the existence of jumps in the system, most of data points fall within 3 standard
deviations away from 0. This figure intensively indicates that the variations (volatility) of
difference of Xt+1 and Xt for every level of Xt are almost same, which means it is reasonable
to assume that σ(·) is a constant function.


1.3 Jump-Diffusion Modeling Procedures
By regularities observed in Section 2, we can specify our model under the so-called “equiva-
lent martingale measure” Q (see, e.g., Duffie (1996)) as follows:

(M1). We assume the volatility function σ(·) is a constant function, namely

σ(x) = σ, x≥0 (1.5)

(M2). By (O2) in Section 2, we assume the instantaneous mean function µ(x) is an affine
function,
µ(x) = A(¯ − x),
x x ≥ 0, (1.6)

where x stands for long-term mean of the process, and A > 0 is the “speed” of process
¯
back to the long-term mean x. We will explain more about these two parameters.
¯

(M3). We assume the pure jump process Jt is a compound Poisson process independent
of continuous part of Xt and {Wt ; t ≥ 0} although this assumption might not be
necessary. More formally, we assume that the intensity of the Poisson process is a
constant λ and jump sizes are i.i.d with same distribution η. From (O1) in Section 2,
we can assume that η is a normal distribution, with mean 0, and standard deviation σJ
although the normality assumption on jump sizes might not be appropriate due to its
lack of fat-tail (One can assume that it follows a double exponential as in Kou (2002)
or Tsay (2002, 2005, Section 6.9)).

By assumptions (M1)-(M3) above, we can reformulate (1.3) as follows:

dXt = A(¯ − Xt )dt + σdWt + dJt ,
x X0 = x0 , (1.7)

where the compensator measure of Jt , ν satisfies:
λ e2
ν(de, dt) = exp − 2
dedt, (1.8)
2
2πσJ 2 σJ
and
EQ [dWt dJs ] = 0, s, t ≥ 0. (1.9)

Using the Itˇ lemma for semi-martingale, we can solve the equation (1.7) explicitly. That
o
is, for any given times t and T (we always assume t ≤ T in the following), we have,
T T
XT = Xt e−A(T −t) + x 1 − e−A(T −t) + e−A(T −t) σ
¯ eA(s−t) dWs + eA(s−t) dJs . (1.10)
t t


By taking the expectation on both sides of (1.10), we obtain

EQ [XT ] = EQ [Xt ]e−A(T −t) + x 1 − e−A(T −t) .
¯ (1.11)

Since A > 0, when T − t → ∞, the first term on the right side of (1.11) will diminish to 0,
while EQ [XT ] → x with exponential rate A. These facts tell us why x is called “long-term
¯ ¯
mean” and A is called the “speed” of process back to the long-term mean.

Suppose that the times of observations of the process are equally-spaced, namely, we
assume we observe the process at the regular times to observe data (Xt1 , Xt2 , . . . , XtN+1 ),
(for notational simplicity, we will denote Xn = Xtn for 1 ≤ n ≤ N + 1 ), where the equal
time interval is defined as ∆ = tn+1 − tn . Then (X1 , X2 , . . . , XN +1 ) follows an AR(1) model;
that is,
Xn+1 = a + bXn + εn+1 , 1 ≤ n ≤ N, (1.12)

where
a = x(1 − e−A∆ ),
¯ b = e−A∆ , (1.13)

and
∆ ∆
εn ∼ σe−A∆ eAs dWs + e−A∆ eAs dJs i.i.d. (1.14)
0 0
Using (1.12), (1.13) and (1.14), to overcome “curse of dimensionality” for estimating pro-
cedure, we propose the so called “two-stage” estimating technique to obtain preliminary
estimate for parameters. Formally speaking, we first estimate parameters A and x by using
¯
Weighted Least Square method, then use residuals to implement MLE estimating procedure
to estimate λ, σ and σJ . So only thing left we need to do is to find the probability density
function for εn , which is given by the following,
 
−λ∆
e x
fεn (x) = φ 
σ 2 σ 2
(1 − e−2A∆ ) (1 − e−2A∆ )
2A 2A
∞ ∆ ∆
e−λ∆ λk 1
+ ...
k! 0 0 σ2
(1 − e−2A∆ ) + k −2A(∆−sl ) σ 2
k=1
2A l=1 e J
 
x
×φ   ds1 . . . dsk , (1.15)
σ2 k −2A(∆−sl ) σ 2
2A
(1 − e−2A∆ ) + l=1 e J

x 2
where φ(x) = √1 e− 2 , namely, the p.d.f of standard normal distribution.
2π


The two-stage estimate for parameters then can be numerically implemented. To estimate
parameters more efficiently, we shall use the whole MLE procedure using Newton-Raphson
algorithm with Two-Stage estimate as initial point of algorithm. Our Two-stage estimates
(based on daily) are as follows:

A = 0.03110101 x = 3.743758
¯ σ = 0.01841 λ = 0.06385 σJ = 0.09299. (1.16)

Our whole MLE estimates (based on daily) are as follows:

A = 0.017124 x = 3.73213
¯ σ = 0.018181 λ = 0.064548 σJ = 0.092432 (1.17)

Now we turn to testing whether the jump diffusion model is adequate. For testing pa-
rameters, we only do test for λ. Equivalently, it tests whether there are jumps for swap rates’
evolution. Remaining parameters’ test can be done similarly. This statistical hypothesis can
be formulated as follows:

H0 : λ = 0 versus H1 : λ > 0. (1.18)

We use the likelihood ratio method to test this hypothesis. It is well known that 2 times
the difference of two maximum log likelihoods converges asymptotically to a χ2 -distribution
with degree of freedom equal to difference of dimensions of two parameter spaces. In this
hypothesis, the degree of freedom is 2 since λ = 0 makes σJ irrelevant to the process. We
find that p-value of test statistic is much less than 0.001, which means that H0 is rejected.
So a model for this dataset without jump could be inappropriate.

1.4 Pricing American-style Options Using Stratifica-
tion Simulation Method
To price an American option by using a simulation method (see, e.g., Glasserman, 2004),
it is always approximated by reducing the American option with intrinsic infinite exercise
opportunities into a “Bermudan” option with finite exercise opportunities. Suppose the ap-
proximated “Bermudan” option can be exercised only at a fixed set of exercise opportunities
t1 < t2 < . . . < tm , which are often equally spaced, and underlying process is denoted by
{Xt ; t ≥ 0}. To reduce notation, we write Xti as Xi . Then, if {Xt } is a Markov process,
{Xi ; 0 ≤ i ≤ m} is a Markov chain, where X0 denotes an initial state of the underlying. Let
hi denote the payoff function for exercise at ti , which is allowed to depend on i. Let Vi (x)


denote the value of the option at ti given Xi = x. By assuming the option has not previ-
ously been exercised, we are ultimately interested in V0 (X0 ). This value can be determined
recursively as follows:
Vm (x) = hm (x) (1.19)

and
Vi−1 (x) = max hi−1 (x), EQ [Di−1,i (Xi )Vi (Xi )|Xi−1 = x] , (1.20)

where i = 1, 2, . . . , m, and Di−1,i (Xi ) stands for the discount factor from ti−1 to ti , which
could have the form as
ti
Di−1,i (Xi ) = exp − r(u)du . (1.21)
ti−1

So for simulation, the main job will be on implementing (1.20), and main difficulty is also
at here. Actually, if the underlying state is of one dimension, for instance in our setting,
then we can efficiently implement (1.20) by stratification method. That is, we discretize not
only time-dimension but also state space. Formally speaking, for each exercise date ti , let
Ai1 , . . . , Aibi be a partition of the state space of Xi into bi subsets. For the initial time 0,
take b0 = 1 and A01 = {X0 }. Define transition probabilities

pi = PQ (Xi+1 ∈ Ai+1,k |Xi ∈ Aij )
j,k (1.22)

for all j = 1, . . . , bi , k = 1, . . . , bi+1 , and i = 0, . . . , m − 1. (This is taken to be 0 if
P Q (Xi ∈ Aij ) = 0.) For each i = 1, . . . , m and j = 1, . . . , bi , we also define

hi,j = EQ [hi (Xi )|Xi ∈ Aij ] (1.23)

takeing this to be 0 if P Q (Xi ∈ Aij ) = 0. Now we consider the backward induction
bi+1

Vij = max hij , pi Vi+1,k
jk (1.24)
k=1

for all j = 1, . . . , bi , k = 1, . . . , bi+1 , and i = 0, . . . , m − 1, initialized with Vmj = hmj . This
method takes the value V01 calculated through (1.24) as an approximation to V0 (X0 ).

To implement this method, we need to do following steps:

(A1) Simulate a reasonably large number of replications of the Markov chain X0 , X1 , . . . , Xm .

i
(A2) Record Njk , the number of paths that move from Aij to Ai+1,k , for all i = 0, . . . , m − 1,
j = 1, . . . , bi and k = 1, . . . , bi+1 .


(A3) Calculate the estimates

pi = Nj,k /(Nj,1 + . . . + Nj,bi )
j,k
i i i
(1.25)

taking the ratio to be 0 whenever the denominator is 0. And calculate hi,j as the
average value of h(Xi ) over those replications in which Xi ∈ Aij , taking it to be 0
whenever there is no path in which Xi ∈ Aij .

(A4) Set Vmj = hmj for all j = 1, . . . , bm , and recursively calculate
bi+1

Vij = max hij , pi Vi+1,k
jk (1.26)
k=1

for all j = 1, . . . , bi , and i = 0, . . . , m − 1. Then V01 just be our estimate of V01 .

For our example, the American-style option is defined with payoff function 1000000(exp(X)−
K)+ , where K = 44 bps, maturity T = 1 year, and initial price exp(X0 ) = 44 bps. Using
parameters presented on (1.17), we simulate 10000 paths with m = 400 exercise opportuni-
ties and decompose state-space into bi = 100 subsets. Then using the above algorithm, we
can approximate the American option price. Based on the simulation with 25 replications,
the mean value and standard deviation of approximates assuming that risk-free interest is
2.5% annually are as follows:

P = 781.762, and sP = 2.632. (1.27)

Note that the estimated value of price based on this jump model is quite close to the real
value.

1.5 Hedging Issues
In the previous implementation, we have assumed risk-free interest is 2.5% annually, and
parameters as presented in (1.17). In this Section, we consider hedging problems given these
parameters. We only discuss first order hedging for the American option. Denote P (S0 )
the option price, where we have omitted all other parameters except initial swap rate in the
∂P
function of P (·). To do first hedging for the derivative is to find the value of ∂S
(S0 ). We can
find this value numerically by using the Euler approximation, namely using first difference
ratio to approximate the partial derivative,
∂P P (S0 + ∆S) − P (S0 )
(S0 ) ≃ (1.28)
∂S ∆S


Then we can use simulation method to find P (S0 + ∆S) for sufficiently small ∆S so that we
can find an approximate “hedging ratio”. For our example, we let ∆S = ±0.25 bps. The
simulated “hedging ratio” is 0.2515. We can use similar technique to find other “Greek”s.

1.6 Conclusions
We present a whole procedure of modelling, estimating, pricing, and hedging for a real
data under a simple jump-diffusion setting. Some shortcomings are obvious in our setting,
since we don’t consider some issues which maybe important for the price of this option.
For instance, we assume interest rate is deterministic, and intensity λ of “jump event” is
constant. Most critically, we don’t deal with the issue observed on (O4) presented on Section
2. But, sometimes we need to compromise between accuracy and tractability in practice,
since calibrating a jump-diffusion model usually needs a large scale of calculating efforts. A
reasonable extension (still no touch on (O4)) to our model is to take so called “multi-factor”
models, which usually include interest rates, CPI, GDP growth rate, volatility, and other
economic variables as factors. See Duffie, Pan, Singleton (2000) for more details.

1.7 References
Cai, Z. and Y. Hong (2003). Nonparametric methods in continuous-time finance: A selective
review. In Recent Advances and Trends in Nonparametric Statistics (M.G. Akritas and
D.M. Politis, eds.), 283-302.

Duffie, D. (2001). Dynamic Asset Pricing Theory, 3th Edition. Princeton University Press,
Princeton, NJ.

Duffie, D., J. Pan and K. Singleton (2000). Transform analysis and asset Pricing for affine
jump-diffusion. Econometrica, 68, 1343-1376.

Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer-Verlag,
New York.

Kou, S.G. (2002). A jump diffusion model for option pricing. Management Science, 48,
1086-1101.

Merton, R.C. (1976). Option pricing when underlying stock return are discontinuous.
Journal of Financial Economics, 3, 125-144.

Stanton, R. (1997). A nonparametric model of term structure dynamics and the market
price of interest rate risk. Journal of Finance, 52, 1973-2002.


Taylor, S. (2005). Asset Price Dynamics, Volatility, And Prediction. Princeton University
Press, Princeton, NJ. (Chapter 4)

Tsay, R.S. (2005). Analysis of Financial Time Series, 2th Edition. John Wiley & Sons,
New York.

Chapter 2

Basic Concepts of Prices and Returns

2.1 Introduction
Any empirical analysis of the dynamics of asset prices through time requires price data which
raise some of the questions:

1. The first question is where we can find the data. There are many sources of data includ-
ing web sites, commercial vendors, university research centers, and financial markets.
Here are some of them, listed below:

(a) CRSP: http://www.crsp.com (US stocks)

(b) Commodity Systems Inc: http://www.csidata.com (Futures)

(c) Datastream: http://www.datastream.com/product/has/ (Stocks, bonds, curren-
cies, etc.)

(d) IFM (Institute for Financial Markets): http://www.theifm.org (futures, US s-
tocks)

(e) Olsen & Associates: http://www.olsen.ch (Currencies, etc.)

(f) Trades and Quotes DB: http://www.nyse.com/marketinfo (US stocks)

(g) US Federal Reserve: http://www.federalreserve.gov/releases (Currencies, etc.)

(h) Yahoo! (free): http://biz.yahoo.com/r/ (Stocks, many countries)

(i) For downloading the Chinese financial data, please see the file on my home page
http://www.math.uncc.edu/˜ zcai/finance-data.doc which is downloadable.

13

CHAPTER 2. BASIC CONCEPTS OF PRICES AND RETURNS 14

Further, the high frequency data (tick by tick data) can be downloaded from the
Bloomberg machine located at Room 33 of Friday Building on our campus but you
might ask Department of Finance for a help. Finally, you can download some data
through the web site at Wharton Research Data Services (WRDS)
http://wrds.wharton.upenn.edu/index.shtml, which our UNCC subscribes partially.
To log in WRDS, you need to have an account which can be obtained by contacting
Jon Finn through e-mail jcfinn@uncc.edu or phone (704) 687-3156.

2. The second question is what the frequency of data is. It depends on what kind of
data you have and what kind of topics you are doing. For study of microstructure of
financial market, you need to have high frequency data. For most of studies, you might
need daily/weekly/monthly data.

3. The third one is how many periods (say, years) (the length) of data we need for analysis.
Theoretically, the larger sample size would be better but it might have structural
changes for a long sample period. In other words, the dynamics might change over
time.

4. The last one is how many prices for each period we wish to obtain and what kind of
price we need.

Answer: It depends on the purpose of your study.

2.2 Basic Definitions
First, we introduce some basic concepts, which you might be very familiar with.

2.2.1 Time Value of Money
Consider an amount $V invested for n years at a simple interest rate of r per annum (where
r is expressed as a decimal). If compounding takes place only at the end of the year, the
future value after n years is:
F Vn = V × (1 + r)n .

If interest is paid m times per year then the future value after n years is:

m r m×n
F Vn = V × (1 + ) .
m


Table 2.1: Illustration of the Eﬀects of Compounding:
The Time Interval Is 1 Year and the Interest Rate is 10% per Annum.
Number Interest rate
Type of payments per period Net Value
Annual 1 0.1 $1.10000
Semiannual 2 0.05 $1.10250
Quarterly 4 0.025 $1.10381
Monthly 12 0.0083 $1.10471
Weekly 52 0.1/52 $1.10506
Daily 365 0.1/365 $1.10516
Continuously ∞ exp(0.1) $1.10517

As m, the frequency of compounding, increases the rate becomes continuously compounded
and it can be shown that the future value becomes:

c r m×n
F Vn = lim V × (1 + ) = V × exp(r × n), (2.1)
m→∞ m
where exp(·) is the exponential function.

Example: Assume that the interest rate of a bank deposit is 10% per annum and the initial
deposit is $1.00. If the bank pays interest once a year, then the net value of the deposit
becomes $1(1+0.1)=$1.1 one year later. If the bank pays interest semi-annually, the 6-month
interest rate is 10%/2 = 5% and the net value is $1(1 0.1/2)2=$1.1025 after the ﬁrst year.
In general, if the bank pays interest m times a year, then the interest rate for each payment
is 10%/m and the net value of the deposit becomes $1(1 0.1/m)m one year later. Table 2.1
gives the results for some commonly used time intervals on a deposit of $1.00 with interest
rate 10% per annum. In particular, the net value approaches $1.1052, which is obtained by
exp(0.1) and referred to as the result of continuous compounding.

2.2.2 Assets and Markets
Financial Assets:

1. Zero-Coupon Bond (discount bond). A zero-coupon bond with maturity date T pro-
vides a monetary unit at date T . At date t with t ≤ T , the zero-coupon bond has a
residual maturity of H = T − t and a price of B(t, H) (or B(t, T − t)), which is the


price at time t,

 (1 + r)−(T −t) pay at the end of maturity day
B(t, T ) = (1 + r/m)−m(T −t) compounding with frequency m

exp(−r(T − t)) continuous compounding

where r is the interest rate. In particular, B(0, T ) is the current, time 0 of the bond,
and B(T, T ) = 1 is equal to the face value, which is a certain amount of money that
the issuing institute (for example, a government, a bank or a company) promises to
exchange the bond for.

2. Coupon Bond. Bonds promising a sequence of payments are called coupon bonds. The
price pt at which the coupon bond is traded at any date t between 0 and the maturity
date T diﬀers from the issuing price p0 .

3. Stocks

4. Buying and Selling Foreign Currency

5. Options

6. More · · ·

For more details about bonds, see the book by Capi´ ski and Zastawniak (2003).
n

2.2.3 Financial Theory
Basic theoretical concepts in ﬁnancial theory (The best book for this aspect is the book by
Cochrane (2002)):

1. Equilibrium Models. (CAPM, CCAPM, market microstructure theory). Our focus
is only on the CAPM. Please read the paper by Cai and Kuan (2008) and the references
therein on the recent developments in the conditional CAP/APT models. Also, for the
market microstructure theory, please read Chapter 3 of Campbell, Lo and MacKinlay
(1997, CLM hereafter), or Part IV of Taylor (2005), or Chapter 5 of Tsay (2005).

2. Absence of Arbitrage Opportunity. The theory is based on the assumption that it
is impossible to achieve sure, strictly positive, gain with a zero initial endowment. This
assumption suggests imposing deterministic inequality restrictions on asses prices.


3. Actuarial Approach. This approach assumes a deterministic environment and em-
phasizes the concept of a fair price of financial asset.

Example: The price of stock at period 0 that provides future dividends d1 , d2 , . . . , dt at
predetermined dates 1, 2, . . . , t has to coincide with the discounted sum of future cash flows:
∞
S0 = dt B(0, t),
t=1

where B(0, t) is the price of the zero-coupon bond with maturity t (discount factor). The
actuarial approach is not confirmed by empirical research because it does not take into
account uncertainty.

2.3 Statistical Features
2.3.1 Prices
Prices: closing prices in stock market; currency exchange rates; option prices; more, · · ·.

2.3.2 Frequency of Observations
It depends on the data available and the questions that interest a researcher. The price
interval between prices should be sufficient to ensure that trade occurs in most intervals
and it is preferable that the volume of trade is substantial. Daily data are fine for most of
the applications. Also, it is important to distinguish the price data indexed by transaction
counts from the data indexed by time of associated transactions.

2.3.3 Definition of Returns
The statistical inference on asset prices is complicated because asset price might have non-
stationary behavior (upward and downward movements). One can transform asset prices
into returns, which empirically display more stationary behavior. Also, returns are scale-free
and not limited to the positiveness. You may notice the difference in the behavior of price
data and returns by looking at IBM prices and IBM returns in Figure 2.1 and Figure 2.2.

1. Return of a financial asset (stock) with price Pt at date t that produces no divi-
dends over the period (t, t + H) is defined as:
Pt+H − Pt
r(t, t + H) = (2.2)
Pt


The stock price of IBM, weekly observations

140

120
Close

100

80

60

40

03
98

00

01
97

99

03
01
99

02

5/
4/

3/

5/
7/

7/

0/
8/
9/

3/

/2
/0

/2

/0
/2

/1

/2
/2
/0

/1

11
06

06

11
09

10

03
02
02

07
Date
The stock price of IBM, monthly observations
140

120
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Figure 2.1: The weekly and monthly prices of IBM stock.

Very often, we will investigate returns at a ﬁxed unitary horizon. In this case H = 1
and return is deﬁned as:
Pt+1 − Pt Pt+1
r(t, t + 1) = = − 1. (2.3)
Pt Pt
Returns r(t, t + H) and r(t, t + 1) in (2.2) and (2.3) are sometimes called the simple
net return. Very often, r(t, t + 1) is simply denoted as rt+1 . The simple gross return is


defined as:
Pt+H
R(t, t + H) = = 1 + r(t, t + H).
Pt
Pt+H Pt+H Pt+H−1 Pt+1
Since Pt
= Pt+H−1 Pt+H−2
× ... × Pt
the R(t, t + H) can be rewritten as:

Pt+H Pt+H−1 Pt+1
R(t, t + H) = × ... ×
Pt+H−1 Pt+H−2 Pt
= R(t + H − 1, t + H) × R(t + H − 2, t + H − 1) × . . . × R(t, t + 1)
H
= R(t + H − j, t + H + 1 − j).
j=1

The simple gross return over H periods is the product of one period returns.

The formula in (2.3) is often replaced by the following approximation:

Pt+1
r(t, t + 1) ≡ rt+1 ≈ ln(Pt+1 ) − ln(Pt ) = ln = ln(R(t, t + 1)). (2.4)
Pt

The return in (2.4) is also known as continuously compounded return or log return. To
see why r(t, t + 1) is called the continuously compounded return, take the exponential
of both sides of (2.4) and rearranging we get

Pt+1 = Pt exp(r(t, t + 1)). (2.5)

By comparing (2.5) with (2.1) one can see that r(t, t + 1) is the continuously com-
pounded growth rate in prices between months t − 1 and t. Rearranging (2.4) one can
show that:
H
r(t, t + H) = r(t + H − j, t + H + 1 − j).
j=1

2. Return of a financial asset (stock) with price Pt at date t that produces dividends
Dt+1 over the period (t, t + 1) is defined as:
Pt+1 + Dt+1 − Pt Pt+1 − Pt Dt+1
r(t, t + 1) = = + , (2.6)
Pt Pt Pt
where Dt+1 /Pt is the ratio of dividend over price (d-p ratio), which is a very important
financial instrument for studying financial behavior.

3. Spot currency returns. Suppose that Pt is the dollar price in period t for one unit
of foreign currency (say, euro). Let i∗ be the continuously compounded interest rate
t−1


The returns of IBM, weekly observations

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The returns of IBM, monthly observations
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Figure 2.2: The weekly and monthly returns of IBM stock.

for deposits in foreign currency from time t − 1 until time t. Then one dollar used to
buy 1/Pt−1 euros in period t − 1, which are sold with accumulated interest in period t,
gives proceeds equal to Pt ∗ exp(i∗ )/Pt−1 and the return is
t−1

rt = log(Pt ) − log(Pt−1 ) + i∗ = pt − pt−1 + it−1 .
t−1
∗

In practice, the foreign interest rate is ignored because it is very small compared with
the magnitude of typical daily logarithmic price change.


4. Futures returns. Suppose Ft,T is the futures price in period t for delivery or cash set-
tlement in some later period T . As there are no dividend payouts on futures contracts,
the futures return is defined as:

rt = log(Ft,T ) − log(Ft−1,T ) = ft,T − ft−1,T ,

where ft,T = log(Ft,T ).

5. Excess return is defined as the difference between the asset’s return and the return
on some reference asset. The reference asset is usually assumed to be riskless and in
practice is usually a short-term Treasury bill return. Excess return is defied as:

z(t, t + 1) = zt+1 = r(t, t + 1) − r0 (t, t + 1), (2.7)

where r0 (t, t + 1) is the reference return from period t to period t + 1.

2.4 Stylized Facts for Financial Returns
When you have data, the first and very important step you need to do is to explore primarily
the data. That is; before you build models for the given data, you need to examine the data to
see what kind of key features the data have, to avoid the mis-specification, so that intuitively,
you have some basic ideas about the data and possible models for the given data. Here are
three important and common properties that are found in almost all sets of daily returns
obtained from a few years of prices:

1. The distribution of returns is not normal (do you believe this?), but it has
the following empirical properties:

• Stationarity. There are two definitions: weakly (second moment) stationary and
strictly stationary. The former is referred in most of applications. Question: How
to check stationarity?

• It is approximately symmetric. Sample estimates of skewness (µ3 /σ 3 , where
µi is the ith central moment µi = E(rt −µ)i , µ is the mean, and σ 2 is the variance)
for daily US stock returns tend to be negative for stock indices but close to zero
or positive for individual stocks.


• It has fat tails. Kurtosis (the ratio of the forth central moment over square of
the second central moment minus 3; that is, γ = µ4 /µ2 − 3) for daily US stock
2

returns are large and positive for both indices and individual stocks which means
that returns have more probability mass in the tail areas than would be predicted
by a normal distribution (leptokurtic or γ > 0).

• It has a high peak. See Figure 2.3 for IMB daily returns by a comparison with
the standard norm.

Figures 2.3-2.4 compare empirical estimates of the probability distribution function

80
Standardized IBM returns

60

40

20

0
−6 −4 −2 0 2 4 6

0.5
Analytical pdf
Empirical pdf
0.4

0.3

0.2

0.1

0
−6 −4 −2 0 2 4 6

Figure 2.3: The empirical distribution of standardized IBM daily returns and the pdf of stan-
dard normal. Notice fat tails of empirical distribution compared with the tails of standard
normal.

(pdf) of standardized IBM and Microsoft (MSFT) returns, zt = (rt − r)/σ, with the
¯
probability density distribution of normal distribution. This empirical density estimate


100
Standardized MSFT returns
80

60

40

20

0
−6 −4 −2 0 2 4 6

0.5
Analytical pdf
Empirical pdf
0.4

0.3

0.2

0.1

0
−6 −4 −2 0 2 4 6

Figure 2.4: The empirical distribution of standardized Microsoft daily returns and the pdf
of standard normal. Notice fat tails of empirical distribution compared with the tails of
standard normal.

has been calculated using nonparametric kernel density estimation:
T
1 1 z − zt
f (z) = K , (2.8)
T t=1
h h

where K(·) is a kernel function and h = h(T ) → 0 as T → ∞ is called bandwidth.
In practice, h = c T −0.2 for some positive c dependent on the features of data. Note
that (2.8) is well known in the nonparametric statistics literature. For details, see the
book by Fan and Gijbels (1996). The estimated kurtosis for the standardized IBM
and Microsoft returns is 5.59 and 5.04 respectively (excess kurtosis, γ/ 24/T ). The
fact that the distribution of returns is not normal implies that classical
linear regression models for returns may be not good enough. A satisfactory
probability distribution for daily returns must have high kurtosis and be either exactly


or approximately symmetric. Figure 2.5 displays the quantile-quantile (Q-Q) plots
for the standardized IBM returns (top panel) and the standardized Microsoft returns
(bottom panel). It is evident that the IBM and MSFT returns are not exactly normally
distributed. For more examples, see Table 1.2 (p. 11) and Figure 1.4 (p.19) in Tsay
(2005) or Table 4.6 and Figures 4.1 and 4.2 (pp 70-72) in Taylor (2005).

QQplot for standardized IBM returns
6
Quantiles of Input Sample

4

2

0

−2

−4

−6
−4 −3 −2 −1 0 1 2 3 4
Standard Normal Quantiles

QQplot for standardized MSFT returns
6
Quantiles of Input Sample

4

2

0

−2

−4

−6
−4 −3 −2 −1 0 1 2 3 4
Standard Normal Quantiles

Figure 2.5: Q-Q plots for the standardized IBM returns (top panel) and the standardized
Microsoft returns (bottom panel).

Question 1: How to model the distribution of a return or returns? (A) Parametric
models; (B) Mixture models (see Section 4.8 in Taylor (2005) and Maheu and McCurdy
(2009)); (C) Nonparametric models.

Question 2: How do you know the distribution of a return belong to a particular family?
(A) Informative way to do a model checking using graphical methods, such as Q-Q plot;


(B) Official way is to do hypothesis testing; say Jarque-Bera test and Kolmogorov-
Smirnov tests or other advanced tests, say nonparametric versus parametric tests.

2. There is almost no correlation between returns for different days. Recall that
the correlation between returns τ periods apart is estimated from T observations by
the sample autocorrelation at lag τ :
T −τ
t=1 (rt − r )(rt+τ
¯ − r)
¯
ρτ = T
¯2
t=1 (rt − r )

where r is the sample mean of all T observations. The command acf() in R is the plot
¯
of ρτ versus τ , which is called the ACF plot.

To test H0 : ρ1 = 0, one can use the Durbin-Watson test statistic which is
T T
2 2
DW = (rt − rt−1 ) / rt .
t=2 t=1

Straightforward calculation shows that DW ≈ 2(1 − ρ1 ), where ρ1 is the lag-1 ACF of
{rt }.

Consider testing that several autocorrelation coefficients are simultaneously zero, i.e.
H0 : ρ1 = ρ2 = . . . = ρm = 0. Under the null hypothesis, it is easy to show (see, Box
and Pierce (1970)) that
m
Q=T ρ2 −→ χm .
k
2
(2.9)
k=1

Ljung and Box (1978) provided the following finite sample correction which yields a
better fit to the χ2 for small sample sizes:
m

m
∗ ρ2
k
Q = T (T + 2) −→ χ2 .
m (2.10)
k=1
T −k

Both are called Q-test and well known in the statistics literature. Of course, they are
very useful in applications.

The function in R for the Ljung-Box test is

Box.test(x, lag = 1, type = c("Box-Pierce", "Ljung-Box"))


and the Durbin-Watson test for autocorrelation of disturbances is

dwtest(formula, order.by = NULL, alternative = c("greater","two.sided",
"less"),iterations = 15, exact = NULL, tol = 1e-10, data = list())

which is in the package lmtest.

3. The correlation between magnitudes of returns on nearby days are positive
and statistically signiﬁcant. Functions of returns can have substantial autocorrela-
tions even though returns have very small autocorrelations. Usually, autocorrelations
are discussed for {|rt |λ }, λ = 1, 2. It is a stylized fact that there is positive dependence
between absolute returns on nearby days, and likewise for squared returns. See Section
4.10 in Taylor (2005) and Section 3.5.8.

The autocorrelations of absolute returns are always positive at a lag one day and
positive dependence continues to be found for several further lags. Squared returns
also exhibit positive positive dependence but to a lesser degree. The dependence
in absolute returns may be explained by volatility clustering or regime
switching or nonlinearity. See Section 4.9 in Taylor (2005).

4. Nonlinearity of the Returns Process. For example, Hong and Lee (2003) con-
ducted studies on exchange rates and they found that some of them are predictable
based on nonlinear time series models. There are many ongoing research activities in
this direction. See Chapter 4 in Tsay (2005) and Cai and Kuan (2008). If we have
time, we will spend some time in exploring further on this topic.

2.5 Problems
1. Download weekly (daily) price data for any two stocks, for example, IBM stock (P1t )
for 01/02/62 - 01/15/08 and for Microsoft stock (P2t ) for 03/13/86 - 01/15/2008.

(a) Create a time series of continuously compounded weekly returns for IBM (r1t )
and for Microsoft (r2t ).

(b) Use the constructed weekly returns to construct a series of monthly returns. You
may assume for simplicity that one month consists of four weeks.

Notes econometricswithr

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