Mathematical techniques in finance : tools for incomplete markets /
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| Author / Creator: | Černý, Aleš, 1971- |
|---|---|
| Imprint: | Princeton, N.J. : Princeton University Press, c2004. |
| Description: | xviii, 378 p. : ill. ; 24 cm. |
| Language: | English |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/5777473 |
Table of Contents:
- Preface
- 1. The Simplest Model of Financial Markets
- 1.1. One-Period Finite State Model
- 1.2. Securities and Their Pay-Offs
- 1.3. Securities as Vectors
- 1.4. Operations on Securities
- 1.5. The Matrix as a Collection of Securities
- 1.6. Transposition
- 1.7. Matrix Multiplication and Portfolios
- 1.8. Systems of Equations and Hedging
- 1.9. Linear Independence and Redundant Securities
- 1.10. The Structure of the Marketed Subspace
- 1.11. The Identity Matrix and Arrow-Debreu Securities
- 1.12. Matrix Inverse
- 1.13. Inverse Matrix and Replicating Portfolios
- 1.14. Complete Market Hedging Formula
- 1.15. Summary
- 1.16. Notes
- 1.17. Exercises
- 2. Arbitrage and Pricing in the One-Period Model
- 2.1. Hedging with Redundant Securities and Incomplete Market
- 2.2. Finding the Best Approximate Hedge
- 2.3. Minimizing the Expected Squared Replication Error
- 2.4. Numerical Stability of Least Squares
- 2.5. Asset Prices, Returns and Portfolio Units
- 2.6. Arbitrage
- 2.7. No-Arbitrage Pricing
- 2.8. State Prices and the Arbitrage Theorem
- 2.9. State Prices and Asset Returns
- 2.10. Risk-Neutral Probabilities
- 2.11. State Prices and No-Arbitrage Pricing
- 2.12. Summary
- 2.13. Notes
- 2.14. Appendix: Least Squares with QR Decomposition
- 2.15. Exercises
- 3. Risk and Return in the One-Period Model
- 3.1. Utility Functions
- 3.2. Expected Utility Maximization
- 3.3. Reporting Expected Utility in Terms of Money
- 3.4. Scale-Free Formulation of the Optimal Investment Problem with the HARA Utility
- 3.5. Quadratic Utility
- 3.6. Reporting Investment Potential in Terms of Sharpe Ratios
- 3.7. The Importance of Arbitrage Adjustment
- 3.8. Portfolio Choice with Near-Arbitrage Opportunities
- 3.9. Generalization of the Sharpe Ratio
- 3.10. Summary
- 3.11. Notes
- 3.12. Exercises
- 4. Numerical Techniques for Optimal Portfolio Selection in Incomplete Markets
- 4.1. Sensitivity Analysis of Portfolio Decisions with the CRRA Utility
- 4.2. Newton's Algorithm for Optimal Investment with CRRA Utility
- 4.3. Optimal CRRA Investment Using Empirical Return Distribution
- 4.4. HARA Portfolio Optimizer
- 4.5. HARA Portfolio Optimization with Several Risky Assets
- 4.6. Quadratic Utility Maximization with Multiple Assets
- 4.7. Summary
- 4.8. Notes
- 4.9. Exercises
- 5. Pricing in Dynamically Complete Markets
- 5.1. Options and Portfolio Insurance
- 5.2. Option Pricing
- 5.3. Dynamic Replicating Trading Strategy
- 5.4. Risk-Neutral Probabilities in a Multi-Period Model
- 5.5. The Law of Iterated Expectations
- 5.6. Summary
- 5.7. Notes
- 5.8. Exercises
- 6. Towards Continuous Time
- 6.1. IID Returns, and the Term Structure of Volatility
- 6.2. Towards Brownian Motion
- 6.3. Towards a Poisson Jump Process
- 6.4. Central Limit Theorem and Infinitely Divisible Distributions
- 6.5. Summary
- 6.6. Notes
- 6.7. Exercises
- 7. Fast Fourier Transform
- 7.1. Introduction to Complex Numbers and the Fourier Transform
- 7.2. Discrete Fourier Transform (DFT)
- 7.3. Fourier Transforms in Finance
- 7.4. Fast Pricing via the Fast Fourier Transform (FFT)
- 7.5. Further Applications of FFTs in Finance
- 7.6. Notes
- 7.7. Appendix
- 7.8. Exercises
- 8. Information Management
- 8.1. Information: Too Much of a Good Thing?
- 8.2. Model-Independent Properties of Conditional Expectation
- 8.3. Summary
- 8.4. Notes
- 8.5. Appendix: Probability Space
- 8.6. Exercises
- 9. Martingales and Change of Measure in Finance
- 9.1. Discounted Asset Prices Are Martingales
- 9.2. Dynamic Arbitrage Theorem
- 9.3. Change of Measure
- 9.4. Dynamic Optimal Portfolio Selection in a Complete Market
- 9.5. Summary
- 9.6. Notes
- 9.7. Exercises
- 10. Brownian Motion and Ito Formulae
- 10.1. Continuous-Time Brownian Motion
- 10.2. Stochastic Integration and Ito Processes
- 10.3. Important Ito Processes
- 10.4. Function of a Stochastic Process: the Ito Formula
- 10.5. Applications of the Ito Formula
- 10.6. Multivariate Ito Formula
- 10.7. Ito Processes as Martingales
- 10.8. Appendix: Proof of the Ito Formula
- 10.9. Summary
- 10.10. Notes
- 10.11. Exercises
- 11. Continuous-Time Finance
- 11.1. Summary of Useful Results
- 11.2. Risk-Neutral Pricing
- 11.3. The Girsanov Theorem
- 11.4. Risk-Neutral Pricing and Absence of Arbitrage
- 11.5. Automatic Generation of PDEs and the Feynman--Kac Formula
- 11.6. Overview of Numerical Methods
- 11.7. Summary
- 11.8. Notes
- 11.9. Appendix: Decomposition of Asset Returns into Uncorrelated Components
- 11.10. Exercises
- 12. Dynamic Option Hedging and Pricing in Incomplete Markets
- 12.1. The Risk in Option Hedging Strategies
- 12.2. Incomplete Market Option Price Bounds
- 12.3. Towards Continuous Time
- 12.4. Derivation of Optimal Hedging Strategy
- 12.5. Summary
- 12.6. Notes
- 12.7. Appendix: Expected Squared Hedging Error in the Black--Scholes Model
- 12.8. Exercises
- Appendix A. Calculus
- A.1. Notation
- A.2. Differentiation
- A.3. Real Function of Several Real Variables
- A.4. Power Series Approximations
- A.5. Optimization
- A.6. Integration
- A.7. Exercises
- Appendix B. Probability
- B.1. Probability Space
- B.2. Conditional Probability
- B.3. Marginal and Joint Distribution
- B.4. Stochastic Independence
- B.5. Expectation Operator
- B.6. Properties of Expectation
- B.7. Mean and Variance
- B.8. Covariance and Correlation
- B.9. Continuous Random Variables
- B.10. Normal Distribution
- B.11. Quantiles
- B.12. Relationships among Standard Statistical Distributions
- B.13. Notes
- B.14. Exercises
- References
- Index
