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Quantitative Finance: Its Development, Mathematical Foundations, and Current Scope

ISBN: 978-0-470-45527-2
448 pages
March 2009
Quantitative Finance: Its Development, Mathematical Foundations, and Current Scope (0470455276) cover image
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.

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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 Is the Markowitz Investor Rational?

7.5 Empirical Project #3.

8. Observed Decisions under Risk.

8.1 Evidence about Choices under Risk.

8.1.1 Allais? Paradox.

8.1.2 Prospect Theory.

8.1.3 Preference Reversals.

8.1.4 Risk Aversion and Diminishing Marginal Utility.

8.2 Toward ‘Behavioral’ Finance.

9. Distributions of Returns.

9.1 Some Background.

9.2 The Normal/Lognormal Model.

9.3 The Stable Model.

9.4 Mixture Models.

9.5 Comparison and Evaluation.

10. Dynamics of Prices and Returns.

10.1 Evidence for First-Moment Independence.

10.2 Random Walks and Martingales.

10.3 Modeling Prices in Continuous Time.

10.3.1 Poisson and Compound-Poisson Processes.

10.3.2 Brownian Motions.

10.3.3 Martingales in Continuous Time.

10.4 Empirical Project #4.

11. Stochastic Calculus.

11.1 Stochastic Integrals.

11.1.1 Ito Integrals with Respect to a B.m.

11.1.2 From It^o Integrals to It^o Processes.

11.1.3 Quadratic-Variations of It^o Processes.

11.1.4 Integrals with Respect to It^o Processes.

11.2 Stochastic Differentials.

11.3 Ito’s Formula for Differentials.

11.3.1 Functions of a B.m. Alone.

11.3.2 Functions of Time and a B.m.

11.3.3 Functions of Time and General It^o Processes.

12. Portfolio Decisions over Time.

12.1 The Consumption-Investment Problem.

12.2 Dynamic Portfolio Decisions.

12.2.1 Optimizing via Dynamic Programming.

12.2.2 A Formulation with Additively-Separable Utility.

13. Optimal Growth.

13.1 Optimal Growth in Discrete Time.

13.2 Optimal Growth in Continuous Time.

13.3 Some Qualifications.

13.4 Empirical Project #5.

14. Dynamic Models for Prices.

14.1 Dynamic Optimization (Again).

14.2 Static Implications: The CAPM.

14.3 Dynamic Implications: The Lucas Model.

14.4 Assessment.

14.4.1 The Puzzles.

14.4.2 The Patches.

14.4.3 Some Reflections.

15. Efficient Markets.

15.1 Event Studies.

15.1.1 Methods.

15.1.2 A Sample Study.

15.2 Dynamic Tests.

15.2.1 Early History.

15.2.2 Implications of the Dynamic Models.

15.2.3 Excess Volatility.

PART III: PARADIGMS FOR PRICING.

16. Static Arbitrage Pricing.

16.1 Pricing Paradigms: Optimization vs. Arbitrage.

16.2 The APT.

16.3 Arbitraging Bonds.

16.4 Pricing a Simple Derivative Asset.

17. Dynamic Arbitrage Pricing.

17.1 Dynamic Replication.

17.2 Modeling Prices of the Assets.

17.3 The Fundamental P.D.E.

17.3.1 The Feynman-Kac Solution to the P.D.E.

17.3.2 Working out the Expectation.

17.4 Allowing Dividends and Time-Varying Rates.

18. Properties of Option Prices.

18.1 Bounds on Prices of European Options.

18.2 Properties of Black-Scholes Prices.

18.3 Delta Hedging.

18.4 Does Black-Scholes StillWork?

18.5 American-Style Options.

18.6 Empirical Project #6.

19. Martingale Pricing.

19.1 Some Preparation.

19.2 Fundamental Theorem of Asset Pricing.

19.3 Implications for Pricing Derivatives.

19.4 Applications.

19.5 Martingale vs. Equilibrium Pricing.

19.6 Numeraires, Short Rates, and E.M.M.s.

19.7 Replication & Uniqueness of the E.M.M.

20. Modeling Volatility.

20.1 Models with Price-Dependent Volatility.

20.1.1 The C.E.V. Model.

20.1.2 The Hobson-Rogers Model.

20.2 ARCH/GARCH Models.

20.3 Stochastic Volatility.

20.4 Is Replication Possible?

21. Discontinuous Price Processes.

21.1 Merton’s Jump-Diffusion Model.

21.2 The Variance-Gamma Model.

21.3 Stock Prices as Branching Processes.

21.4 Is Replication Possible?

22. Options on Jump Processes.

22.1 Options under Jump-Diffusions.

22.2 A Primer on Characteristic Functions.

22.3 Using Fourier Methods to Price Options.

22.4 Applications to Jump Models.

23. Options on S.V. Processes.

23.1 Independent Price/Volatility Shocks.

23.2 Dependent Price/Volatility Shocks.

23.3 Adding Jumps to the S.V. Model.

23.4 Further Advances.

23.5 Empirical Project #7.

Solutions to Exercises.

References.

Index.

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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.

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  • Presented as a course in quantitative finance and includes exercises and worked solutions as well as empirical projects that aim to teach the mathematical tools of finance by using them to model the behavior of prices and markets

  • Emphasizes instruction and technique and also covers the essential topics that are typically included in a survey course in quantitative finance: portfolio theory, decision theory, pricing of primary assets, pricing of derivatives, and the empirical behavior of prices

  • Adheres to a self-teaching presentation style.  Math tools are only presented as their applications are required.  Important formulas and derivations are worked out in enough detail so that readers will learn associated techniques as well as results. 

  • Presents material in order of its historical development, which provides readers with a sense as to how the field has progressed
  • Provides exercises at the end of each chapter that progress in difficulty and are designed to both consolidate and extend the topics covered in the text

  • Utilizes MATLAB® for the empirical projects, and complete data sets are available via the book's related FTP site

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