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Fall 2011 (Blog ORDER)

• FINAL EXAM FINAL Version 4.1 (PDF and Latex Source with mydefs) posted 8:45PM 12/5. Obvious typo fixed on problem 5.

• There are twelve problems. Check here for bug reports (or version changes). Send me email if you think you have found a bug.

• Small Typos Fixed: 12/4 6pm Problem 9. Now correct. Earlier misplace tau. No change to version numer.

• Exams Due: MONDAY DEC 19 at NOON.

• Coversheet: Latex Source and PDF

• Typo in 4(b) corrected (11/29 8AM). Typo corrected in 5(b) on (12/1 at 4:30 PM)

• About the Black-Scholes Formula... We'll we've arrived at the derivation of the PDE and the classic formula. We'll leave the classical solution to the reading of the text and work toward a more probabilistic analysis via the Feynman-Kac formula. We'll then circle back, develop the martingale representation theorem and the Girsanov theorem. We'll then use this machinery to give the Harrision-Kreps risk neutral view of contingent claims pricing. We'll be headed for a treatment like that of Janson and Tysk. We'll not aim for the same level of generality, but we'll take the same glide path. [Almost off topic -- a little link on Theory and Proof.]

• In the meanwhile, you might read the vernerable Wikipedia article on the Black-Scholes formula. It's decently balanced. You'll also see that the "derivation" it gives is completely obfuscatory since it does not want to own up to the twin issues of (a) true risk neutrality of the hedeged portfolio and (2) the isssue of whether the hedged portfolio is really self financing. You might also explore some of the links given by the Wikipedia --- especially Wilmoth's piece and the Derman-Talib piece. Finally, the Wikipedia piece on the Greeks is worth reading, though parts need to be taken with a gain of salt.

• Good News! No new HW this week. I'll start dealing out pieces of the final exam pretty soon.

• Jimmy Wang found a serious typo (a thought-oh?) in the last problem of HW No. 7. I wrote f(t,X_t) when I should have simply written f(X_t), without the t variable. II have posted a corrected version of HW7 Due November 2 (PDF and Source).

• Here is one Web resource on Separable ODEs. It's brief but it should be a sufficient reminder if you have seen this technique before. If this solution method is completely new to you, you may want to read the corresponding chapter in some elementary ODE book.

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• Symmetry? Here the bunny makes a hand shaddow. This is a reflection symmery --- in a couple of senses.

• Homework 6 due Wednesday October 26 (PDF and Latex Source). By this time our theory is starting to get pretty rich, so you definitely need to spend a chunk of time doing the reading. Also, I keep the problem sets short, so you may feel the need for more problems: Check out the problems in the text! Those problems are different from the ones I hand out, but the problems in the text still speak to the same issues. In fact, you can often pull a out a good hint for a homework problem if you read through the problems in the book.

• Some hints on problems 2 and 3 of Homework 5. The generic hint is "read the book and your notes." The specific hints are (1) think how the Ito Isometry may help and (2) think how the Doob maximal inequality might help. Also, consider Polya's advice: "Anytime you face a problem, ask--- Have I seen a problem with a similar conclusion? Have I seen a problem with a similar hypothesis? Can I use the result of that old problem or can I use facts associated with the solution of that problem? If you find a copy of Polya's book, "How to Solve It" you should read it!. Although it is written for high school students, it is brillant.

• What is Brownian Motion in n-Dimensions? Here you just take each coordinate locations at time t to be given by independent Brownian motions.

• Homework No 5. is due on Wednesday October 19. (PDF or Latex Source) I will be in Chicago until Sunday, and mostly out of email contact, so if you find a "bug" you'll just need to make a best-judgement decision and forge ahead.

• Naturally, you should continue with your reading. Independent of what we manage to cover in class you should complete you reading of Chapters 6 and 7. The messages of Chaper7 are somewhat "technical" in the sense that you can just skim over them and you will still be on track. You can be the master of your "time allocation." Once we get to Chapter 8, we are working on the "money ball." Chapter 8 is one that everyone should MASTER.

• Victor Mole of Tulane University has assigned himself the selfless task (with helpers) of providing proofs for all the integrals in the compendium of integrals first collected by Gradshteyn and Ryzhik. It is a work in progress (many integrals are done, so don't be put off by the fact that many are missing from the first few "pages."). These proofs are being published in SCIENTIA (a new Chilean journal). Sample article.

• Now this is the work of a great Barista -- (or perhaps photoshop?). Maybe you can have a cup like this as you read Chapters 6 and 7 over our little Fall Beak!

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• There is no homework due in the Week of Oct 10, but you should complete your reading of Chapter 6 and get some head start on Chapter 7. These are not chapters that you really need to master at the moment, but you should at least be informed about what lies behind the "calculus" of stochastic calculus. These chapters also give you some practice with complete spaces, Cauchy sequences, subsequence arguments, and Doob's maximal inequality.

• T. Tao provides free access to a draft of his book on measure theory. It has more about the measure theory of the real line than we will ever need, but it is certainly worth a look. I particularly like his problems solving strategies (page 221 et. seq.). The draft will probably be un-posted at some point, so you may as well snag the pdf while you can do so honestly. My only caution is to not get too lost in the book; it's not quite the measure theory that one needs for probability theory (though kindred, of course.)

• REMINDER: I post the homework a week or more ahead of the time it is due. This gives you a big chunk of time to think about the problems, do the reading, and fill in any missing background. I'll answer any question any time, but if you have a burning question on Tuesday night, then --- with a few special exceptions --- you needed to get started earlier.

• Homework No. 4 (Due 10/5). PDF and Latex Source NOTE: Because of Fall break there is no class on Monday Oct 10 and there will be no homework due in the shortened week of October 10. You should complete your reading of the text through Chapter 5 (omitting Section 5.4). Over the week, you should read Chapter 6. Some parts are "tedious" but learn enough of the detail so you can be honest about understanding the needs for the individual steps leading to the full definition of the Ito Integral.

• I did a paper long ago on "Second Guessing" that is relevant to one of the problems of HW3. You probably don't want to dig too deeply into it, but you may be amused by the introduction and the connection to the HW.

• Comment on HW3: Here we say that two stochastic processes are equivalent if they have the same joint distribution for any finite set of times t_1, t_2, ..., t_n. This is sometimes called "equivalent in distribution." It means that for any event you describe, the "event" will have the same probably for each of the two processes. Here "event" is in quotes since the two processes can live on completely different spaces.

• CHANGE OF PLAN: The add/drop date for graduate students turns out to have been last Friday, September 23. Undergraduates have a last withdrawal of November 18. The good-news-bad-news is that for lack of a , there will be no in-class mid-term.

• Kevin Slavin (High quality video on HF Trading)

• Homework No. 3 (Due 9/ 28). PDF and Latex Source

• Latex Source for No. 1 and No. 2. (Note: 9/14 3:30 pm: Earlier an out-of- date version of this file was posted. Note that the current latex corresponds to the PDF posted above).

• Electronic Journal of the History of Probability and Statistics, Special Issue on Martingales. This is a little "too much like history" for my taste but it does provide information that I have not seen elsewhere.

Is this course for you?

• This course should be useful for well-prepared students who are in the fields of finance, economics, statistics, or mathematics, 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) may not be explicitly assumed, but many function spaces will be introduced and used in the course and students who have not seen these before face heavy sledding.

• "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. Basic mathematical analysis is the core prerequisite, 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.

• 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. Still, never has a MBA had a successful experience with this class; only MBA's with plenty of Ph.D. level quant experience should consider this. Undergraduates have succeed in this class, but it is a very tough challenge; it's only feasible if you have already had 530-531.

• 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 IM'ing, 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. This is a critical step to genuine learning.

• Grading --- There will be an in-class midterm exam that will be given and graded before the drop date. The final will a take-home and it will be longish. It will be a central part of your learning experience. On the final, you can consult any book, but you may not discuss these exam problems with any other person.

• 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