Stochastic Calculus and Financial Applications: (45 Stochastic Modelling and Applied Probability)

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.

Table of 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.

Review :
From the reviews: MATHEMATICAL REVIEWS "…on the whole, the results are presented carefully and thoroughly, and I expect that readers will find that this combination of a careful development of stochastic calculus with many details and examples is very useful and will enable them to apply the whole theory confidently." SHORT BOOK REVIEWS "This is a world of 'lovely exercises' that are 'very good good for the soul', 'honest martingales', 'bedrock approximations', portfolios that are 'born to lose', 'intuitive but bogus arguments', and 'embarrassingly crude insights'. In short, this is a book on stochastic calculus of a different flavour. Intuition is not sacrificed for rigour nor rigour for intuition.The main results are reinforced with simple special cases, and only when the intuitive foundations are laid does the auhtor resort to the formalism of probability. The coverage is limited to the essentials but nevertheless includes topics that will catch the eye of experts (such as the wavelet construction of Brownian motion). This is one of the most interesting and easiest reads in the discipline; a gem of a book." JOURNAL OF THE AMERICAN STOCHASTIC ASSOCIATION "The book is indeed well written, with many insightful comments. I certainly would recommend it to students wishing to learn stochastic calculus and its applications to the Black-Sholes option-pricing theory…I thoroughly enjoyed reading this book. The author is to be complimented for his efforts in providing many useful insights behind the various theories. It is a superb introduction to stochastic calculus and Brownian motion…An interesting feature in this book is its coverage of partial differential equations." "It is clear that this is a fairly comprehensive introduction to the tools of (classical) mathematical finance. … the text has much to offer. … In addition, the writing style is refreshingly informal and makes a book about a rathertechnical subject surprisingly enjoyable to read. In short, despite the recent deluge of textbooks in this area, I know of no better book for self-study." (Christian Kleiber, Statistical Papers, Vol. 46 (2), 2005) "Steele’s book is a sophisticated introduction to stochastic calculus with applications from basic Black-Scholes theory. … I highly recommend the book. His style is wonderful, and concepts really build on one another. … it offers one of the most elegant treatments of the subject that I know of." (www.riskbook.com, May, 2006) "As is clear from the title of this book, it is concerned with applications of stochastic calculus to finance. … one naturally judges the book by three criteria: topic selection, organization, and exposition. In all three domains the book succeeds. The topics selected are rich enough … he or she will benefit from the book. … there are innovations as well … from the pedagogic standpoint." (Philip Protter, SIAM Review, Vol. 43 (4), 2001) "This book offers rich information and a mathematically honest treatment of stochastic calculus and of its use in the theory of finance … . The author gradually builds the reader’s ability to grasp stochastic concepts and techniques … . the author’s presentation of stochastic models in finance and economy is precise and extensive … . Each chapter is accompanied by a collection of rather challenging exercises … ." (EMS Newsletter, December, 2002) "The present book ‘is designed for students who want to develop professional skill in stochastic calculus and its application to problems in finance’. … the textbook … retains a lovely lecture style focusing basic ideas and not formalities and technical details of stochastic processes needed for finance. I can strongly recommend this book to students of mathematics and physics as well as non-experts in probability theory who are interested in stochastic finance." (H. –J. Girlich, Zeitschriftfür Analysis und ihre Anwendungen, Vol. 21 (4), 2002) "The last few years have been a fertile period for books on stochastic calculus and its financial implications, but this one differs from the many mainstream treatments … . The style of the book creates the atmosphere of a lively lecture … . Each chapter ends with a section of carefully chosen exercises, preceded by some motivating remarks. … I really liked the book." (R. Grübel, Statistics & Decisions, Vol. 20 (4), 2002) "This book gives an introduction to stochastic calculus … with applications in mathematical finance. … As the preface says, ‘This is a text with an attitude, and it is designed to reflect, wherever possible and appropriate, a prejudice for the concrete over the abstract’. This is also reflected in the style of writing which is unusually lively for a mathematics book. … on the whole, the results are presented carefully and thoroughly … ." (Martin Schweizer, Zentralblatt MATH, Vol. 962, 2001) "This is a book on stochastic calculus of a different flavour. Intuition is not sacrificed for rigour nor rigour for intuition. The main results are reinforced with simple special cases … . This is one of the most interesting and easiest reads in the discipline; a gem of a book." (D. L. McLeish, Short Book Reviews, Vol. 21 (1), 2001)