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Full Description
This book is designed for students who want to develop professional skill in stochastic calculus and its application to problems in finance. The Wharton School course that forms the basis for this book is designed for energetic students who have had some experience with probability and statistics but have not had ad vanced courses in stochastic processes. Although the course assumes only a modest background, it moves quickly, and in the end, students can expect to have tools that are deep enough and rich enough to be relied on throughout their professional careers. The course begins with simple random walk and the analysis of gambling games. This material is used to motivate the theory of martingales, and, after reaching a decent level of confidence with discrete processes, the course takes up the more de manding development of continuous-time stochastic processes, especially Brownian motion. The construction of Brownian motion is given in detail, and enough mate rial on the subtle nature of Brownian paths is developed for the student to evolve a good sense of when intuition can be trusted and when it cannot. The course then takes up the Ito integral in earnest. The development of stochastic integration aims to be careful and complete without being pedantic.
Contents
1. Random Walk and First Step Analysis.- 1.1. First Step Analysis.- 1.2. Time and Infinity.- 1.3. Tossing an Unfair Coin.- 1.4. Numerical Calculation and Intuition.- 1.5. First Steps with Generating Functions.- 1.6. Exercises.- 2. First Martingale Steps.- 2.1. Classic Examples.- 2.2. New Martingales from Old.- 2.3. Revisiting the Old Ruins.- 2.4. Submartingales.- 2.5. Doob's Inequalities.- 2.6. Martingale Convergence.- 2.7. Exercises.- 3. Brownian Motion.- 3.1. Covariances and Characteristic Functions.- 3.2. Visions of a Series Approximation.- 3.3. Two Wavelets.- 3.4. Wavelet Representation of Brownian Motion.- 3.5. Scaling and Inverting Brownian Motion.- 3.6. Exercises.- 4. Martingales: The Next Steps.- 4.1. Foundation Stones.- 4.2. Conditional Expectations.- 4.3. Uniform Integrability.- 4.4. Martingales in Continuous Time.- 4.5. Classic Brownian Motion Martingales.- 4.6. Exercises.- 5. Richness of Paths.- 5.1. Quantitative Smoothness.- 5.2. Not Too Smooth.- 5.3. Two Reflection Principles.- 5.4. The Invariance Principle and Donsker's Theorem.- 5.5. Random Walks Inside Brownian Motion.- 5.6. Exercises.- 6. Itô Integration.- 6.1. Definition of the Ito Integral: First Two Steps.- 6.2. Third Step: Itô's Integral as a Process.- 6.3. The Integral Sign: Benefits and Costs.- 6.4. An Explicit Calculation.- 6.5. Pathwise Interpretation of Ito Integrals.- 6.6. Approximation in H2.- 6.7. Exercises.- 7. Localization and Itô's Integral.- 7.1. Itô's Integral on L2LOC.- 7.2. An Intuitive Representation.- 7.3. Why Just L2LOC?.- 7.4. Local Martingales and Honest Ones.- 7.5. Alternative Fields and Changes of Time.- 7.6. Exercises.- 8. Itô's Formula.- 8.1. Analysis and Synthesis.- 8.2. First Consequences and Enhancements.- 8.3. Vector Extension and Harmonic Functions.-8.4. Functions of Processes.- 8.5. The General Ito Formula.- 8.6. Quadratic Variation.- 8.7. Exercises.- 9. Stochastic Differential Equations.- 9.1. Matching Itô's Coefficients.- 9.2. Ornstein-Uhlenbeck Processes.- 9.3. Matching Product Process Coefficients.- 9.4. Existence and Uniqueness Theorems.- 9.5. Systems of SDEs.- 9.6. Exercises.- 10. Arbitrage and SDEs.- 10.1. Replication and Three Examples of Arbitrage.- 10.2. The Black-Scholes Model.- 10.3. The Black-Scholes Formula.- 10.4. Two Original Derivations.- 10.5. The Perplexing Power of a Formula.- 10.6. Exercises.- 11. The Diffusion Equation.- 11.1. The Diffusion of Mice.- 11.2. Solutions of the Diffusion Equation.- 11.3. Uniqueness of Solutions.- 11.4. How to Solve the Black-Scholes PDE.- 11.5. Uniqueness and the Black-Scholes PDE.- 11.6. Exercises.- 12. Representation Theorems.- 12.1. Stochastic Integral Representation Theorem.- 12.2. The Martingale Representation Theorem.- 12.3. Continuity of Conditional Expectations.- 12.4. Lévy's Representation Theorem.- 12.5. Two Consequences of Lévy's Representation.- 12.6. Bedrock Approximation Techniques.- 12.7. Exercises.- 13. Girsanov Theory.- 13.1. Importance Sampling.- 13.2. Tilting a Process.- 13.3. Simplest Girsanov Theorem.- 13.4. Creation of Martingales.- 13.5. Shifting the General Drift.- 13.6. Exponential Martingales and Novikov's Condition.- 13.7. Exercises.- 14. Arbitrage and Martingales.- 14.1. Reexamination of the Binomial Arbitrage.- 14.2. The Valuation Formula in Continuous Time.- 14.3. The Black-Scholes Formula via Martingales.- 14.4. American Options.- 14.5. Self-Financing and Self-Doubt.- 14.6. Admissible Strategies and Completeness.- 14.7. Perspective on Theory and Practice.- 14.8. Exercises.- 15. The Feynman-Kac Connection.- 15.1. FirstLinks.- 15.2. The Feynman-Kac Connection for Brownian Motion.- 15.3. Lévy's Arcsin Law.- 15.4. The Feynman-Kac Connection for Diffusions.- 15.5. Feynman-Kac and the Black-Scholes PDEs.- 15.6. Exercises.- Appendix I. Mathematical Tools.- Appendix II. Comments and Credits.
