Quantitative Finance

A rigorous, yet accessible, introduction to essential topics in
mathematical finance

Presented as a course on the topic, Quantitative Finance traces
the evolution of financial theory and provides an overview of core
topics associated with financial investments. With its thorough
explanations and use of real-world examples, this book carefully
outlines instructions and techniques for working with essential
topics found within quantitative finance including portfolio
theory, pricing of derivatives, decision theory, and the empirical
behavior of prices.

The author begins with introductory chapters on mathematical
analysis and probability theory, which provide the needed tools for
modeling portfolio choice and pricing in discrete time. Next, a
review of the basic arithmetic of compounding as well as the
relationships that exist among bond prices and spot and forward
interest rates is presented.? Additional topics covered
include:

• Dividend discount models

• Markowitz mean-variance theory

• The Capital Asset Pricing Model

• Static?portfolio theory based on the expected-utility
  paradigm

• Familiar probability models for marginal distributions of
  returns and the dynamic behavior of security prices

The final chapters of the book delve into the paradigms of
pricing and present the application of martingale pricing in
advanced models of price dynamics. Also included is a step-by-step
discussion on the use of Fourier methods to solve for
arbitrage-free prices when underlying price dynamics are modeled in
realistic, but complex ways.

Throughout the book, the author presents insight on current
approaches along with comments on the unique difficulties that
exist in the study of financial markets. These reflections
illustrate the evolving nature of the financial field and help
readers develop analytical techniques and tools to apply in their
everyday work. Exercises at the end of most chapters progress in
difficulty, and selected worked-out solutions are available in the
appendix. In addition, numerous empirical projects utilize
MATLAB® and Minitab® to demonstrate the mathematical
tools of finance for modeling the behavior of prices and markets.
Data sets that accompany these projects can be found via the book's
FTP site.

Quantitative Finance is an excellent book for courses in
quantitative finance or financial engineering at the
upper-undergraduate and graduate levels. It is also a valuable
resource for practitioners in related fields including engineering,
finance, and economics.

Autorentext

T. W. Epps, PhD, is Professor Emeritus of both Economics and Statistics at the University of Virginia.?A member of the American Finance Association, the American Statistical Association, and the Institute of Mathematical Statistics, Dr. Epps has published numerous journal articles in the areas of statistical theory, financial markets, time series analysis, and econometrics.

Zusammenfassung
A rigorous, yet accessible, introduction to essential topics in mathematical finance Presented as a course on the topic, Quantitative Finance traces the evolution of financial theory and provides an overview of core topics associated with financial investments. With its thorough explanations and use of real-world examples, this book carefully outlines instructions and techniques for working with essential topics found within quantitative finance including portfolio theory, pricing of derivatives, decision theory, and the empirical behavior of prices.

The author begins with introductory chapters on mathematical analysis and probability theory, which provide the needed tools for modeling portfolio choice and pricing in discrete time. Next, a review of the basic arithmetic of compounding as well as the relationships that exist among bond prices and spot and forward interest rates is presented.? Additional topics covered include:

• Dividend discount models

• Markowitz mean-variance theory

• The Capital Asset Pricing Model

• Static?portfolio theory based on the expected-utility paradigm

• Familiar probability models for marginal distributions of returns and the dynamic behavior of security prices

The final chapters of the book delve into the paradigms of pricing and present the application of martingale pricing in advanced models of price dynamics. Also included is a step-by-step discussion on the use of Fourier methods to solve for arbitrage-free prices when underlying price dynamics are modeled in realistic, but complex ways.

Throughout the book, the author presents insight on current approaches along with comments on the unique difficulties that exist in the study of financial markets. These reflections illustrate the evolving nature of the financial field and help readers develop analytical techniques and tools to apply in their everyday work. Exercises at the end of most chapters progress in difficulty, and selected worked-out solutions are available in the appendix. In addition, numerous empirical projects utilize MATLAB® and Minitab® to demonstrate the mathematical tools of finance for modeling the behavior of prices and markets. Data sets that accompany these projects can be found via the book's FTP site.

Quantitative Finance is an excellent book for courses in quantitative finance or financial engineering at the upper-undergraduate and graduate levels. It is also a valuable resource for practitioners in related fields including engineering, finance, and economics.

Inhalt
Preface. PART I: PERSPECTIVE AND PREPARATION.

1. Introduction and Overview.

1.1 An Elemental View of Assets and Markets.

1.1.1 Assets as Bundles of Claims.

1.1.2 Financial Markets as Transportation Agents.

1.1.3 Why Is Transportation Desirable?

1.1.4 What Vehicles Are Available?

1.1.5 What Is There to Learn about Assets and Markets?

1.1.6 Why the Need for Quantitative Finance?

1.2 Where We Go from Here.

2. Tools from Calculus and Analysis.

2.1 Some Basics from Calculus.

2.2 Elements of Measure Theory.

2.2.1 Sets and Collections of Sets.

2.2.2 Set Functions and Measures.

2.3 Integration.

2.3.1 Riemann-Stieltjes.

2.3.2 Lebesgue/Lebesgue-Stieltjes.

2.3.3 Properties of the Integral.

2.4 Changes of Measure.

3. Probability.

3.1 Probability Spaces.

3.2 Random Variables and Their Distributions.

3.3 Independence of R.V.s.

3.4 Expectation.

3.4.1 Moments.

3.4.2 Conditional Expectations and Moments.

3.4.3 Generating Functions.

3.5 Changes of Probability Measure.

3.6 Convergence Concepts.

3.7 Laws of Large Numbers and Central Limit Theorems.

3.8 Important Models for Distributions.

3.8.1 Continuous Models.

3.8.2 Discrete Models.

PART II: PORTFOLIOS AND PRICES.

4. Interest and Bond Prices.

4.1 Interest Rates and Compounding.

4.2 Bond Prices, Yields, and Spot Rates.

4.3 Forward Bond Prices and Rates.

4.4 Empirical Project #1.

5. Models of Portfolio Choice.

5.1 Models That Ignore Risk.

5.2 Mean-Variance Portfolio Theory.

5.2.1 Mean-Variance 'Efficient' Portfolios.

5.2.2 The Single-Index Model.

5.3 Empirical Project #2.

6. Prices in a Mean-VarianceWorld.

6.1 The Assumptions.

6.2 The Derivation.

6.3 Interpretation.

6.4 Empirical Evidence.

6.5 Some Reflections.

7. Rational Decisions under Risk.

7.1 The Setting and the Axioms.

7.2 The Expected-Utility Theorem.

7.3 Applying Expected-Utility Theory.

7.3.1 Implementing EU Theory in Financial Modeling.

7.3.2 Inferring Utilities and Beliefs.

7.3.3 Qualitative Properties of Utility Functions.

7.3.4 Measures of Risk Aversion.

7.3.5 Examples of Utility Functions.

7.3.6 Some Qualitative Implications of the EU Model.

7.3.7 Stochastic Dominance.

7.4 …