Stochastic Calculus and Financial Applications

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.

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)