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

Preface.

PART I: PERSPECTIVE AND PREPARATION.

1. Introduction and Overview.

1.1 An Elemental View of Assets and Markets.

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.

2.3 Integration.

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.

3.5 Changes of Probability Measure.

3.6 Convergence Concepts.

3.7 Laws of Large Numbers and Central Limit Theorems.

3.8 Important Models for Distributions.

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.

5.3 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.

7.4 Is the Markowitz Investor Rational?

7.5 Empirical Project #3.

8. Observed Decisions under Risk.

8.1 Evidence about Choices under Risk.

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.

10.4 Empirical Project #4.

11. Stochastic Calculus.

11.1 Stochastic Integrals.

11.2 Stochastic Differentials.

11.3 Ito's Formula for Differentials.

12. Portfolio Decisions over Time.

12.1 The Consumption-Investment Problem.

12.2 Dynamic Portfolio Decisions.

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.

15. Efficient Markets.

15.1 Event Studies.

15.2 Dynamic Tests.

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.

17.4 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.

20.2 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.

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.