Although there are many books on mathematical finance, few deal with the statistical aspects of modern data analysis as applied to financial problems. This book fills this gap by addressing some of the most challenging issues facing any financial engineer. It shows how sophisticated mathematics and modern statistical techniques can be used in concrete financial problems. Concerns of risk management are addressed by the control of extreme values, the fitting of distributions with heavy tails, the computation of values at risk (VaR), and other measures of risk. Data description techniques such as principal component analysis (PCA), smoothing, and regression are applied to the construction of yield and forward curve. Nonparametric estimation and nonlinear filtering are used for option pricing and earnings prediction. The book is intended for undergraduate students majoring in financial engineering, or graduate students in a Master in finance or MBA program. Because it was designed as a teaching vehicle, it is sprinkled with practical examples using market data, and each chapter ends with exercises. Practical examples are solved in the computing environment of S-PLUS. They illustrate problems occurring in the commodity and energy markets, the fixed income markets as well as the equity markets, and even some new emerging markets like the weather markets. The book can help quantitative analysts by guiding them through the details of statistical model estimation and implementation. It will also be of interest to researchers wishing to manipulate financial data, implement abstract concepts, and test mathematical theories, especially by addressing practical issues that are often neglected in the presentation of the theory. Rene Carmona is the Paul M. Wythes '55 Professor of Engineering and Finance at Princeton University in the department of Operations Research and Financial Engineering and Director of Graduate Studies of the Bendheim Center for Finance. His publications include over seventy articles and six books in probability and statistics. He was elected Fellow of the Institute of Mathematical Statistics in 1984, and he is on the editorial board of several peer-reviewed journals and book series. Professor Carmona has developed computer programs for teaching of statistics, for research in signal analysis, and more recently, he contributed the library EVANESCE for the analysis of heavy tail distributions and copulas. The latter was included in the latest version of S-Plus. He has worked for many years on energy and weather derivatives, and he is recognized as a leading researcher and consultant in this area.
From the reviews: As can be seen from the chapters' contents, the breadth of topics covered of this book is impressive. Overall, this is a very nice book for introducing students to a variety of models for analyzing financial data." Journal of Statistical Software, June 2004 "The author, a fellow of the Institute of Mathematical Statistics, presents a solid dose of theory and methodology." Technometrics, May 2005 "This book is a text for an undergraduate course in data analysis focused on financial applications. It is not an S-Plus book but rather covers the main problems arising in data analysis techniques in financial engineering..As the book is based on lectures for a course on statsitical analysis of financial data, a trade off between the depth at which the toics are presented and the computational implementations are kept in balance. This textbook will be very helpful for a general course in financial engineering." The American Statistician, November 2005 "This textbook appears to be primarily intended as an introduction to statistical analysis of financial data ... . the book provides the reader with a practical computational approach to financial analytical techniques. It should appeal to instructors who prefer an applied-based text to a theoretical one. I enjoyed the use of simulation based illustrations and will be using some of the ideas in the future. The book could be used for teaching a third-year undergraduate or post-graduate (honours level), course in a statistics department or in a program designed for finance." (Gary D Sharp, SASA News, March, 2006) "S-plus, a popular software for statisticians, has many books devoted to teach it. ... the book would be very good choice as a lab manual providing many useful rules of thumb. ... the book doubtlessly provides a pleasant introduction to statistics using S-plus. The friendly tone throughout certainly adds to the charm. Simple yet detailed exercises at the end of each chapter offer a gentle massage for the brain." (Arnab Chakraborty, Sankhya, Vol. 66 (3), 2004) "This is an excellent text, written by a well known expert in the field, dealing with statistical analysis of financial data. ... As remarked by the author, the emphasis of the book is on graphical and computational methods for the analysis of financial data. ... The book is clearly written and remarkably free of typos. I believe it will be a very useful addition to the existing books and I highly recommend it." (Pedro A. Morettin, Zentralblatt MATH, Vol. 1055, 2005) "This is a timely book on modern data analysis with a difference: the examples and applications are predominantly taken from Finance Engineering. ... This book will help fill a statistical gap in the otherwise heavily theoretical literature in mathematical finance." (D. L. McLeish, Short Book Reviews, Vol. 24 (2), 2004) "The seven chapters are an excellent resource to anyone wishing to learn more about the application of statistics to financial data. ... A comprehensive reference section is given and the book has the S-PLUS codes that are needed to perform the statistical modelling. ... The reference section is extremely useful and comprehensive. Libraries should be encouraged to purchase copies of this text for undergraduate and post-graduate students in finance and statistics." (Isaac Dialsingh, Significance, Vol. 3 (3), 2006)
This is the first book at the graduate textbook level to discuss analyzing financial data with S-PLUS. Its originality lies in the introduction of tools for the estimation and simulation of heavy tail distributions and copulas, the computation of measures of risk, and the principal component analysis of yield curves. The book is aimed at undergraduate students in financial engineering; master students in finance and MBA's, and to practitioners with financial data analysis concerns.
Contents
Part I Data Exploration, Estimation And Simulation
1 Univariate Exploratory Data Analysis
1.1 Data, Random Variables and Their Distributions
1.1.1 The PCS Data
1.1.2 The S&P 500 Index and Financial Returns
1.1.3 Random Variables and Their Distributions
1.1.4 Examples of Probability Distribution Families
1.2 First Exploratory Data Analysis Tools
1.2.1 Random Samples
1.2.2 Histograms
1.3 More Nonparametric Density Estimation
1.3.1 Kernel Density Estimation
1.3.2 Comparison with the Histogram
1.3.3 S&P Daily Returns
1.3.4 Importance of the Choice of the Bandwidth
1.4 Quantiles and Q-Q Plots
1.4.1 Understanding the Meaning of Q-Q Plots
1.4.2 Value at Risk and Expected Shortfall
1.5 Estimation from Empirical Data
1.5.1 The Empirical Distribution Function
1.5.2 Order Statistics
1.5.3 Empirical Q-Q Plots
1.6 Random Generators and Monte Carlo Samples
1.7 Extremes and Heavy Tail Distributions
1.7.1 S&P Daily Returns, Once More
1.7.2 The Example of the PCS Index
1.7.3 The Example of the Weekly S&P Returns
Problems
Notes & Complements
2 Multivariate Data Exploration
2.1 Multivariate Data and First Measure of Dependence
2.1.1 Density Estimation
2.1.2 The Correlation Coefficient
2.2 The Multivariate Normal Distribution
2.2.1 Simulation of Random Samples
2.2.2 The Bivariate Case
2.2.3 A Simulation Example
2.2.4 Let's Have Some Coffee
2.2.5 Is the Joint Distribution Normal?
2.3 Marginals and More Measures of Dependence
2.3.1 Estimation of the Coffee Log-Return Distributions
2.3.2 More Measures of Dependence
2.4 Copulas and Random Simulations
2.4.1 Copulas
2.4.2 First Examples of Copula Families
2.4.3 Copulas and General Bivariate Distributions
2.4.4 Fitting Copulas
2.4.5 Monte Carlo Simulations with Copulas
2.4.6 A Risk Management Example
2.5 Principal Component Analysis
2.5.1 Identification of the Principal Components of a Data Set
2.5.2 PCA with S-Plus
2.5.3 Effective Dimension of the Space of Yield Curves
2.5.4 Swap Rate Curves
Appendix 1: Calculus with Random Vectors and Matrices
Appendix 2: Families of Copulas
Problems
Notes & Complements
Part II Regression
3 Parametric Regression
3.1 Simple Linear Regression
3.1.1 Getting the Data
3.1.2 First Plots
3.1.3 Regression Set-up
3.1.4 Simple Linear Regression
3.1.5 Cost Minimizations
3.1.6 Regression as a Minimization Problem
3.2 Regression for Prediction & Sensitivities
3.2.1 Prediction
3.2.2 Introductory Discussion of Sensitivity and Robustness
3.2.3 Comparing L2 and L1 Regressions
3.2.4 Taking Another Look at the Coffee Data
3.3 Smoothing versus Distribution Theory
3.3.1 Regression and Conditional Expectation
3.3.2 Maximum Likelihood Approach
3.4 Multiple Regression
3.4.1 Notation
3.4.2 The S-Plus Function lm
3.4.3 R2 as a Regression Diagnostic
3.5 Matrix Formulation and Linear Models
3.5.1 Linear Models
3.5.2 Least Squares (Linear) Regression Revisited
3.5.3 First Extensions
3.5.4 Testing the CAPM
3.6 Polynomial Regression
3.6.1 Polynomial Regression as a Linear Model
3.6.2 Example of S-Plus Commands
3.6.3 Important Remark
3.6.4 Prediction with Polynomial Regression
3.6.5 Choice of the Degree p
3.7 Nonlinear Regression
3.8 Term Structure of Interest Rates: A Crash Course
3.9 Parametric Yield Curve Estimation
3.9.1 Estimation Procedures
3.9.2 Practical Implementation
3.9.3 S-Plus Experiments
3.9.4 Concluding Remarks
Appendix: Cautionary Notes on Some S-Plus Idiosyncracies
Problems
Notes & Complements
4 Local & Nonparametric Regression
4.1 Review of the Regression Setup
4.2 Natural Splines as Local Smoothers
4.3 Nonparametric Scatterplot Smoothers
4.3.1 Smoothing Splines
4.3.2 Locally Weighted Regression
4.3.3 A Robust Smoother
4.3.4 The Super Smoother
4.3.5 The Kernel Smoother
4.4 More Yield Curve Estimation
4.4.1 A First Estimation Method
4.4.2 A Direct Application of Smoothing Splines
4.4.3 US and Japanese Instantaneous Forward Rates
4.5 Multivariate Kernel Regression
4.5.1 Running the Kernel in S-Plus
4.5.2 An Example Involving the June 1998 S&P Futures Contract 193
4.6 Projection Pursuit Regression
4.6.1 The S-Plus Function ppreg
4.6.2 ppreg Prediction of the S&P Indicators
4.7 Nonparametric Option Pricing
4.7.1 Generalities on Option Pricing
4.7.2 Nonparametric Pricing Alternatives
4.7.3 Description of the Data
4.7.4 The Actual Experiment
4.7.5 Numerical Results
Appendix: Kernel Density Estimation & Kernel Regression
Problems
Notes & Complements
Part III Time Series & State Space Models
5 Time Series Models: Ar, Ma, Arma, & All That
5.1 Notation and First Definitions
5.1.1 Notation
5.1.2 Regular Time Series and Signals
5.1.3 Calendar and Irregular Time Series
5.1.4 Example of Daily S&P 500 Futures Contracts
5.2 High Frequency Data
5.2.1 TimeDate Manipulations
5.3 Time Dependent Statistics and Stationarity
5.3.1 Statistical Moments
5.3.2 The Notion of Stationarity
5.3.3 The Search for Stationarity
5.3.4 The Example of the CO2 Concentrations
5.4 First Examples of Models
5.4.1 White Noise
5.4.2 Random Walk
5.4.3 Auto Regressive Time Series
5.4.4 Moving Average Time Series
5.4.5 Using the Backward Shift Operator B
5.4.6 Linear Processes
5.4.7 Causality, Stationarity and Invertibility
5.4.8 ARMA Time Series
5.4.9 ARIMA Models
5.5 Fitting Models to Data
5.5.1 Practical Steps
5.5.2 S-Plus Implementation
5.6 Putting a Price on Temperature
5.6.1 Generalities on Degree Days
5.6.2 Temperature Options
5.6.3 Statistical Analysis of Temperature Historical Data
Appendix: More S-Plus Idiosyncracies
Problems
Notes & Complements
6 Multivariate Time Series, Linear Systems & Kalman Filtering
6.1 Multivariate Time Series
6.1.1 Stationarity and Auto-Covariance Functions
6.1.2 Multivariate White Noise
6.1.3 Multivariate AR Models
6.1.4 Back to Temperature Options
6.1.5 Multivariate MA & ARIMA Models
6.1.6 Cointegration
6.2 State Space Models
6.3 Factor Models as Hidden Markov Processes
6.4 Kalman Filtering of Linear Systems
6.4.1 One-Step-Ahead Prediction
6.4.2 Derivation of the Recursive Filtering Equations
6.4.3 Writing an S Function for Kalman Prediction
6.4.4 Filtering
6.4.5 More Predictions
6.4.6 Estimation of the Parameters
6.5 Applications to Linear Models
6.5.1 State Space Representation of Linear Models
6.5.2 Linear Models with Time Varying Coefficients
6.5.3 CAPM with Time Varying â's
6.6 State Space Representation of Time Series
6.6.1 The Case of AR Series
6.6.2 The General Case of ARMA Series
6.6.3 Fitting ARMA Models by Maximum Likelihood
6.7 Example: Prediction of Quarterly Earnings
Problems
Notes & Complements
7 Nonlinear Time Series: Models and Simulation
7.1 First Nonlinear Time Series Models
7.1.1 Fractional Time Series
7.1.2 Nonlinear Auto-Regressive Series
7.1.3 Statistical Estimation
7.2 More Nonlinear Models: ARCH, GARCH & All That
7.2.1 Motivation
7.2.2 ARCH Models
7.2.3 GARCH Models
7.2.4 S-Plus Commands
7.2.5 Fitting a GARCH Model to Real Data
7.2.6 Generalizations
7.3 Stochastic Volatility Models
7.4 Discretization of Stochastic Differential Equations
7.4.1 Discretization Schemes
7.4.2 Monte Carlo Simulations: A First Example
7.5 Random Simulation and Scenario Generation
7.5.1 A Simple Model for the S&P 500 Index
7.5.2 Modeling the Short Interest Rate
7.5.3 Modeling the Spread
7.5.4 Putting Everything Together
7.6 Filtering of Nonlinear Systems
7.6.1 Hidden Markov Models
7.6.2 General Filtering Approach
7.6.3 Particle Filter Approximations
7.6.4 Filtering in Finance? Statistical Issues
7.6.5 Application: Tracking Volatility
Appendix: Preparing Index Data
Problems
Notes & Complements
Appendix: An Introduction To S And S-plus
References
Notation Index
Data Set Index
S-Plus Index
Author Index
Subject Index