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What's New

• In the Fall 2008 course, Chapter 1 on random walk will be skipped. In the past I used it for motivation, but we really do get the same motivation if we move more quickly to the study of martingales. I would encourage students to read the chapter, and I will distribute copies of the solutions of the problems. Chapter 1 has a lot of value for people who need either a review of probability or a little boost up the the 'graduate level' in probability.
• I expect to devote a little less time to the very detailed and formal development of the Ito integral. We will still put energy into this and students will definitely get a serious appreciation of the details of the construction, but we won't twist and noodle every detail.
• I plan to spend a lot more time with Girsanov Theory, and I will take a different line of development than the one I used for the original text. This will let us spend considerable time dealing with both the mechanics and the intellectual issues of the "risk neutral measure." The reason for this shift is that the Harrison-Kreps view of option pricing is the one that 'has legs'. All of the current theory is done (at least conceptually) from the Harrison-Kreps perspective.

Is this course for you?

• I believe that this is a useful course for well-prepared students in finance, economics, and statistics, but it is definitely directed toward students who also have a genuine interest in fundamental mathematics. Naturally, we deal with financial theory to a serious extent, but, in this course, financial theory and financial practice are the salad and desert --- not the main course.
• Our work requires a high level of comfort with the tools of real analysis, including uniform continuity, Cauchy's convergence criterion, notions of integrability, and calculations in inner product spaces. Knowledge of function spaces (L-one, L-two, Hilbert space, etc) is not explicitly assumed, but many function spaces will be introduced and used in the course.
• Measure-theoretic probability theory enters the conversation regularly, but, with a reasonable amount of work, it can also be picked up as the course progresses. Still, there is no denying that some knowledge of measure theory will be useful --- at least to the level of having understood the Borel-Cantelli lemmas, the definition of convergence with probability one, and the Dominated Convergence Theorem. These can be learned from the Appendix of the book, but it may be useful to read the little book on integration by Bartle.
• Students who have had Statistics 530-531 are perfectly well prepared, as are students with a graduate course in real analysis. Many students with lesser backgrounds have taken the course and done well. It is substantially a matter of priorities and motivation.
• We are after the absolute core of stochastic calculus, and we are going after it in the simplest way that we can possibly muster. If we are honest at each turn, this challenge is plenty hard enough.
• There is a syllabus for 955 but this page is the place to come for up-to-date information about the course content and procedures.

Course Policies

• House Keeping --- Please no cell phones, no open lap tops, no newspapers, no hoagies, no Au Bon Pain Salad Boxes, etc. A coffee or a soft drink is OK, but please be kind to your neighbor --- we have a bounded space.
• Homework --- It is important to solve problems and to discuss the solutions of problems. We will do a lot of this in class, and I will also suggest problems whose solutions will be posted on the web site. Since individuals vary so much in the effort that they can commit to a graduate course, these regular "homework problems" will not be graded. Nevertheless, they are critical to genuine learning.
• Grading --- There will be a midterm exam and a final exam. Each of these will be take-home exams. They will be longish. They will be a central part of your learning experience. You can consult any book, but you may not discuss these exam problems with any other person.
• In Statistics 956 we have an entirely different process (teams, etc), but for Statistics 955 this is the chosen design. Naturally, you have lots of opportunity to discuss the non-graded homeworks with anyone you choose. Hey, with me for example!
• Auditing --- Certainly. You are most welcome.

Course Topics

On the first day of class, I will draw a mind map that puts the topics of the course into a frame that I believe to be much more meaningful than a simple list. The picture is based on a rectangle with vertices: a=martingales b=Brownian motion c=Ito calculus d=arbitrage . The rest of our topics hang naturally off of these vertices.

What the "big picture" does not show directly --- but which I try to underscore at every turn --- is the importance of problem solving. There really are "techniques for solving problems," and one finds a different "place to stand" once even a modest mastery of these techniques has been attained.

Random walks and first step analysis
First martingale steps
Brownian motion
Martingales: The next steps
Richness of paths
Itô integration
Localization and Itô's integral
Itô's formula
Stochastic differential equations
Arbitrage and SDEs
The diffusion equation
Representation theorems
Girsanov theory
Arbitrage and martingales
The Feynman-Kac connection

The Fall 2006 Blog

• In the Fall of 2006, we stepped over Chapter 1 (Random Walk) lightly, but we carefully covered all of the material of Chapter 2 (Discrete Time Martingales) except prehaps for the martingale limit theorems. In particular, we covered Doob's maximal inequalities with honest completeness. These inequalities are foundation stones.
• Next on the agenda is Chapter 3 Brownian Motion, where we will also just hit the highlights such as construction of Brownian motion. In passing, we'll take the opportunity to review notions like orthonormality; this will be needed when we discuss the Ito isometry. We won't dig into path properties of BM except at a "cultural levels," but we will get all of the " moment formulas" for BM. They will appear in many subsequent computations.
• We've completed the construction of Brownian motion, and we are ready to look at martingales in continuous time. We'll solve some concrete problems to confirm that you really have already learns some useful stuff.
• We'll also discussed the tools that are needed to obtain Doob's stopping time theorem in continuous time. At one level, the theorem is obvious by analogy with discrete time, and it is tempting to say no more. Still, an hones proof requires the introduction of several technical ideas --- especially uniform integrability. We'll compromise between the two extreme paths of "pure analogy" and "complete" proof. If someone wants the complete proof, we'll have done enough in class to make it accessible.
• We've now competed the construction of the Ito integral, including the extension that takes us into the land of local martingales.
• Ito's Formula is now in our tool kit in many different forms, including those that use the Box calculus. The mid-term exam is out.
• We've go on to harvest all the returns on our investments. In particular we have developed the martingale theory of arbitrage pricing. To make this happen we learned more about Ito's formula, developed the martingale representation theorem, learned about Levy's characterization of Brownian motion, and developed "Novikov Theory" --- including the criteria of Kazamaki and Krylov.
• The Final exam has been posted, and --- as I now write (Dec, 2006)--- even handed in. You can now look over the solutions if you have a moment.

Final Exam --- Solutions, Comments

I've posted the solutions to the final, just in case you get curious. Also, so far I have graded about half of the exams, and they are looking good. Several people have made big improvements over their performance on the midterm, and this is a delight to me. I won't be in the office until January, but feel free to stop by the office then and pick up your exam. It is a violation of University rules to put the exams where there is open access, though those messy MBA Profs break this rule every year.

Please do check out the Stat 956: Financial Time Series. It's of a much different character than SCFA. It is substantially empirical, and it looks at a wide range of financial assets. Thanks for participating in SCFAn and I hope to see many of you in the spring class!

FINAL When's It Due?

Just like it says in the university schedule, your exam is due on Tuesday December 19 at 2PM. I will be around from 1:30 to receive it in person. By 2PM you should simply hand in whatever you have. Don't stress yourself out about extra time. This exam will have been in your hands for more than 28 days, the gestation period for a mouse. No matter what, you should let go of it at the due date. Also, please use the required cover sheet --- a small revision has been posted .

The the of "two problems called No. 12" has been fixed. Please use the current (and obvious) numbering for your solutions. Also, the one missing parenthesis has been added. Both of these correction were mentioned in class on Wednesday Nov 22, but you still may want to print out the current version of the exam.

Final Is PosTed--- AnD IT's LOVELY!

Today is the Wednesday before Thanksgiving. This is traditionally a "play day" at Universities around the United States. We'll do our part. We'll celebrate with games --- juggling --- open loops --- Yor's Lemma --- and --- best of all --- an introductory review of the final exam problems. Many are easy, all are beautiful. Just take a look!

Black-Scholes --- Retrospective

Donald MacKenzie has written an excellent history of the Black-Scholes formula, its evolution, and its impact on the development of options markets. He adds many professional details to the shop stories that I have told in class. This is as close to full story as one is likely to get, and, for me, it is a delight to read.

Please Review The Solutions

The solutions to the midterm all all now carefully checked, but it is always possible for new errors to creep in. PLEASE let me know if you suspect a bug in the any of the midterm solutions. Among other things, this will help me grade correctly! Also, no matter if you did well on the midterm or poorly, you can probably learn a lot by carefully working though the solutions.