21-260: Final Wrap Up

Well... All the exams have been graded, and final grades have been recorded. I've posted your final averages on the blackboard site. In most cases Blackboard seems to calculate the same average I do. (Blackboard sometimes seem to insist on doing it's own thing.) One exception is for those of you who have turned in a late written homework assignment. The Blackboard average does not account for any penalties which may apply.

You can see the grade cutoffs for the final exam, as well as for your final averages, on this page of statistics for the course.

21-260: Exam #3 Solutions

The Blackboard link for the Exam #3 solutions now points to the correct file. Sorry for the mistake earlier.

Office Hours for finals week

My office hours this week will be in Wean Hall 6201 or a nearby comfortable location at

Tuesday 6:00 - 7:00 pm
Wednesday 12:30 - 1:30 pm, 6:00 - 7:00 pm
Thursday 11:30am - 1:00pm

unfortunately I have a couple of finals of my own on Tuesday, so I won't be able to meet with people during the morning. I may, possibly, perhaps, have the most recent homework graded by Tuesday evening. It will definitely be done by my office hours on Wednesday.

Chris

21-260: Final Exam

I've posted a Review Page for the final exam. The Final Exam is cumulative. The linked review page only covers the material since Exam #3. You should review the material on the review pages for Exam #1, Exam #2 and Exam #3 to ensure you are prepared for the Final Exam.

UPDATE: You can ignore problems #9 and #11b on the "Old Exam Problems" from the Final Exam Review Page. The Heat Equation will not be covered.

21-260: Final Exam

Our final exam will be given on Thursday, December 10 from 5:30-8:30pm in UC McConomy. Students who have a Legitimate Conflict may take a makeup exam on Friday, December 11 from 1:00-4:00pm in UC McConomy (with students taking their 21-127 exam).

By "Legitimate Conflict" I mean (1) two exams scheduled at the same time, (2) three consecutive exams, (3) a University sponsored trip, or (4) other situations that I deem to be roughly similar in gravity to these. You must contact me in advance if you wish to take the makeup exam.

Office Hours Changed, Week 15

This Thursday my office hours are moved to 7:00 - 8:00 pm for personal reasons.

21-260: Weeks #14 and #15

The reading and homework assignments are now posted through the end of the semester. There is an online assignment only for Week #14, and both written and online assignments for Week #15. Both online assignments are due Wednesday, December 2 (during Week #15) and the written assignment is due Friday, December 4, the last day of classes.

Office Hours Cancelled

I will not be holding any office hours this week due to the Thanksgiving break. (obviously)

Office Hours Changed, Week 13

In view of the exam being on Wednesday and no homework on Friday, I'm moving my Thursday office hours to Tuesday. So I'll hold office hours 8:30-9:30 am, 12:30-1:00 pm, and 6:00-7:00 pm on Tuesday with none on Thursday.

21-260: Exam #3

I've posted a Review Page for Exam #3. I will be holding a reivew session on Monday Evening, from 7:00-8:30 in DH 2315.

UPDATE: I've added to the review page the table of Laplace transforms you will be given with Wednesday's exam. I've also corrected the numbering of the sections in Chapter 3. The exam covers all the material we looked at from Chapter 3, and Sections 6.1 and 6.2 from Chapter 6.

Global Entrepreneurship Week

Next Week is Global Entrepreneurship Week.

Preparing for Finals Workshop

The end of the semester is drawing near. Are you . . . .

• Feeling anxious about final exams?


• Worried that you'll work hard but not make the grade you
  expect?


• Facing a mountain of material to review for your exams?

If any of these apply to you then you could benefit from the Preparing for
Finals Workshop on

Wednesday, November 11
6:30-8:00pm
Cyert B6-B

Sign up today for your opportunity to maximize your final exam preparation!

Register online at http://www.cmu.edu/acadev/
Or contact Academic Development for more information

21-260: Week #12

The reading and homework assignments for Week #12 have been posted. You can follow the link from the Schedule page.

21-260: Week #11

The reading and homework assignments for Week #11 have been posted. You can follow the link from the Schedule page.

levity

I will Derive

21-260: Week #10

The assignments for Week #10 have been posted. You can follow the link from the Schedlue Page.

Academic Presentations

Academic Development is holding a series of two workshops on giving academic presentations. You can register online:

Dates: Thursdays -- Oct. 29 & Nov. 5
Time: 5:00pm to 6:30pm
Location: Cyert Hall B6-A

21-260: Week #9

I've posted the reading and assignments for Week #9. Unlike most weeks, there is a written assignment due on Wednesday and no assignment due on Friday. Exam #2 will be given on Friday. A review page for Exam #2 has been posted. You can follow the links from the Schedule page.

21-260: Week #8

The reading and homework assignments for Week #8 have been posted. You can follow the link from the Schedule page.

UPDATE: I've corrected an error on the Wiley-PLUS assignment for this week. I removed the problem I mistakenly assigned from Section 7.9 and replaced it with a problem from Section 7.8.

UPDATE: There is no written assignment due this week. There was some wording on the assignment page that seemed to contradict this, but I've changed that now.

21-260: Exam #1 Solutions

I've posted solutions to the first mid-term exam. They are in the Course Documents section of the Blackboard site.

question 7.27 part c

Just to clarify what was said in class today, I said to rewrite the equation

c1*u + c2*v = 0

into

a*(something_1) + b*(something_2)=0

Since something_1 and something_2 are both functions of t (a variable), you can pick any t_0 (a constant) and plug it for t so that something_1 and something_2 become constants. I mentioned that you should make a smart choice for t_0 so algebraically the problem becomes simpler. The important thing is that you can pick ANY VALUE for t_0. After making some choice for t_0, if you find that c1=c2=0, then it follows that u and v are linearly independent.

21-260: Week #7

The reading and homework assignments for Week #7 have been posted. You can follow the link from the Schedule page.

21-260: Online assignment #6

The Online Assignment for Week #6 is due at 11:00pm tonight. When I set up the assignment, I neglected to set the time, so it is listed as 9:00am. When I went back into the system to change that today, I got the warning that "gradebook information might be lost." I decided not to risk it! Go ahead and complete the assignment by 11:00pm as usual. It will be marked as late, but I'll sort it out later.

Thursday Office Hour Changes

I will be changing my office hours for some upcoming Thursdays to accommodate the Graduate Student Seminar this week and my trip to the Circus in two weeks. The hours will be:

Oct. 1: 6:35pm - 7:35pm
Oct. 15: 2:00pm - 3:00pm

All other Thursdays will remain unchanged.

21-260: Week #6

THe Reading and Homework assignments for Week #6 have been posted. You can follow the link from the Schedule page.

Presentation Points

Each homework problem is graded on a scale of 7 points for correctness and 3 points for presentation. The purpose of grading on presentation is to encourage students to write solutions to problems in a way that another person who had not done the problem themselves could understand the solution. This is an extremely important skill, not just for mathematicians but in all walks of life where mathematical conclusions need to be explained to supervisors, clients, inspectors, and such.

But it can be pretty subjective if we just say it's a score for "presentation". So I'll explain what my grading philosophy on this is.

All differential-equations-related steps are to be identified in words. Examples of this would be attempting to separate variables, multiplying the equation by an integrating factor, or plugging in the initial conditions to solve for a constant of integration. You will learn other techniques during this course that will have to be identified similarly.

Some calculus steps should also be identified in words, such as u-substitution or trig-substitution to solve an integral, or differentiating to find a maximum or minimum point. You don't have to explain that the derivative of sine is cosine, or that you can pull a constant out of an integral.

Very few algebra or arithmetic steps need to be identified. Finding the roots of a polynomial or doing a partial fraction decomposition are the only ones I can think of off the top of my head.

So this is how points will be awarded:

3: All the above steps are identified using words.
2: A significant proportion of the steps that should be identified using words are not, but it is immediately clear to me what the student is doing.
1: A significant proportion of the steps that should be identified using words are not, and it takes some effort on my part to comprehend what the student is doing, due to minor steps being omitted, extremely unclear or incorrect notation, or the work being scattered around the page rather than flowing from top to bottom and right to left.
0: Vital steps are completely missing from the solution, or the problem is not attempted at all. Note that this may also affect the correctness score. I also reserve the right to give 0 points in special cases at my discretion, for instance if a problem requiring significant mathematical work is answered with an incorrect solution that requires little or no work. I don't want to give out zeros, trust me.

If a problem has multiple parts, with different presentation quality among the parts, the score will be a rough average of the presentation score for each part.

Following the above simple presentation rules gives all students the opportunity to get points that may make up for minor errors in the solutions themselves. Please take advantage of that opportunity.

21-260: Exam#1 Wrapup

Well, Exam #1 has been graded and (mostly) returned to you. The grade cutoffs for the exam are 75 (for an A), 65 (B), 55 (C) and 45 (D). To compute grade cutoffs for your mid-semester and final grades, I'll average the cutoffs for each assignment. That way, for instance, someone who scores just above the A cutoff on each assignment will have a final average that is just above the A cutoff.

I've taken the time to tabulate some statistics for the course, including the current grade cutoffs for cumulative averages.

Section E recitation cancelled, Thursday Office Hours

Just a reminder: since the recitation for section E at 12:30 has been canceled due to G-20 preparations, I will be holding office hours from 12:30-1:20 today, so you can pick up your exams there.

semi-stable points

Before I mentioned that in order to determine if a point is stable, unstable, or semistable you can take the derivative of f(y) and plug the critical points into f '(y). Well, if for some "y" you get that f '(y) = 0, this does not mean that the critical point is semistable, it simply means that you can't use this derivative test, and you have to check the intervals around that critical point. However, if f '(y) > 0, then the point is unstable and if f '(y) < 0, then the point is stable.

Office Hours

Ravi will be holding extra office hours this Sunday from 12 Noon - 1 PM in Wean 6215. You will be able to pick up the homework (Section B), and then I'll probably leave it with the other TA's.

Strange Office Hours, or, How I Learned To Stop Worrying And Love Differential Equations

In view of the upcoming exam on Monday, I will be holding bonus office hours (and missing the Vikings-Lions game) this coming Sunday from 1:00pm to 3:00pm.

21-260: Euler's Method in Dfield

This week's written assignment asks you to use dfield to plot numerical solutions generated using Euler's method. Euler's method is not the default method in dfield, but it can easily be changed.

First, plot the direction field for the equation you want to study. Then in the DFIELD Direction Field Window go to the Options menu. Choose Options > ODE Solver > Euler. This brings up the Dfield Solver Settings Window with a text box to enter the step size. Enter your desired step size and click "Apply". (If you don't click Apply, there will be no change to the step size. This is, for some reason, annoyingly frustrating. But maybe that is just me.)

In order to gain more precision in plotting solutions to initial value problems, you may wish to enter them from the keyboard, rather than clicking in the window. To do this, select Solution > Keyboard Input of Initial Value in the DFIELD Direction Field Window. This brings up the DFIELD Keyboard Input window, which performs just as you would expect.

21-260: Exam #1

The Review page for Exam #1 has been posted. You can follow the link from the Schedule Page.

21-260: Week #4

The reading and homework assignmetns for Week #3 have been posted. You can follow the link from the Schedule page.

21-260: Week #3

The reading and homework assignmetns for Week #3 have been posted. You can follow the link from the Schedule page.

Office Hours

My Office Hours will be on Tuesdays from 10:30-11:30 in Wean 6215.

21-260: Week #2

The reading and homework assignments for Week #2 have been posted. The assignment is linked from the schedule page.

21-260 Recitation Sections

I've had a number of people approach me regarding issues with their recitation section, and wait lists. The important thing for the semester is that you register for the recitation section you will be attending. I know many of you have said that your preferred section is full.

Most of the wait lists have dwindled down to only a few students. Here is what I would like you all to do in order to get everyone registered for the course:

1. 
   If you are registered for a section that you can not attend, switch your registration to a section that you can attend.
   



2. 
   If you can not register for a section that you can attend, put yourself on a wait list for a section that you can attend.
   

I understand that you may be reluctant to give up your spot in a section to put yourself on a wait list, but I don't foresee that any student will be unable to take the course due to enrollment limits. If it continues to be an issue, we'll work something out.


UPDATE: As I said above, the important thing is to register for the recitation you will be attending. It is better if you attend the lecture that corresponds to your recitations section, but if that is not possible, you can certainly attend the other lecture. They are, in principle, identical.

The Father of Sinh

A student asked me about the origin of the hyperbolic functions in one of my recitations today. The Wikipedia article credits the Alsatian mathematician-physicist-philosopher-spelunker Johann Heinrich Lambert (who was apparently obsessed with equal-area map projections) with the invention of these functions, but his own article doesn't mention them at all. Maybe a disgruntled calculus student edited them out.

Google Timeline lists a reference back in 1631, by an English mathematician Richard Norwood, but that link is actually referring to trigonometric functions. The second reference in Timeline (A History of Mathematical Notations: Notations Mainly in Higher Mathematics by Florian Cajori) says that the hyperbolic functions were first used by Vincenzo Riccati in 1757, but he used the notation "Sh." for hyperbolic sine and "Ch." for hyperbolic cosine. Lambert is said to have invented the notation that we now use for these functions (sinh,cosh,tanh) in 1768. However, several other notations were used for these functions well into the 1900s.

So, we can take away from this that

(a) Wikipedia is not to be trusted, even in math;
(b) mathematical notation is all over the place; and
(c) inventing good notation for a concept will make you as famous as the people who invented the actual concept.

Minor correction for 21-260-A recitation 8/27

This morning, on problem 19, I broke up the derivative of t^r into two cases (r = 0 and r ≠ 0). I later figured out that this is actually unnecessary, since we were assuming that t > 0 in that problem and the others in that section of the homework. r*t^(r-1) = 0 when r = 0, and this is the correct derivative of a constant. (There would be a problem here if t were allowed to be 0, since we'd be raising 0 to a negative power, which is not allowed.)

So, you don't have to treat the case where r = 0 and r = 1 separately in that problem, which should make it a bit less tedious.

Also note that the equation in problem #1 was actually linear (I mistakenly listed it as nonlinear at the beginning of recitation). A linear differential equation may have terms that are not linear in t, as long as all terms are linear in y and its derivatives.

21-260: Welcome to 21-260

Welcome to 21-260 Differential Equations. As I mentioned in class this morning, there are several things your should be aware of for the course. First of all, the course website. This will be the source of much of your information about the course, via the schedule page and the link to this blog. Also, the Blackboard site, where you can look at your recorded grades, and find homework and exam solutions, and the Wiley-PLUS site, where you will complete online homework.

If you need to purchase a registration code for the Wiley-PLUS site, you can do that here. Once you have registered with the Wiley-PLUS syste, you can access the course materials via this link.

UPDATE: The course-specific URL you need to register/access the Wiley-PLUS page is http://edugen.wiley.com/edugen/class/cls128224/

Did you ever wonder...

... why we have a moon? Differential Equations can help explain.

An Interview with James Stewart

The MAA has posted an interesting interview with James Stewart, the author of the calculus textbook we use here at Carnegie Mellon. Included: his passion for music, and his remarkable home, "Integral House."

21-270: Final Wrapup

Well, the final grades for the course have been turned in. You can look at your exam grade in the Blackboard site, which will remain up for a little while longer. I'm not sure how long. Here are the final statistics for the course, including the cutoffs for final averages.

Have a great summer!

21-270: Final Exam

The Final Exam will be given on Thursday morning, from 8:30-11:30am. The exam will be in McConomy Auditorium. I will try to schedule a reveiw session for Wednesday afternoon, and I'll post information about it here when I know more.

I've posted a review page with some information about the final. Note that the Final will be cumulative. You should be sure to review the study problems from the first three exams, as well as your homework and exams.

UPDATE: Solutions to Exam #2 are now available from the blackboard site.

UPDATE: I've posted the formula sheet for the final exam on the blackboard site and the review page.

UPDATE: Solutions to most of Exam #3 are now available on the blackboard site.

UPDATE: Here are the current Statistics for the course.

UPDATE: Grade cutoffs have been corrected on the Statistics page.

Who are the Monsters?

Recently I received an email from the AAUP (American Association of University Professors), as I do from time to time. The text of the email is reproduced here. It begins as follows

Over 40 years ago, when I was still an undergraduate at Antioch College, the student government sent out a large number of letters to controversial or accomplished Americans and invited them to talk on campus. One who accepted was George Lincoln Rockwell, head of the American Nazi Party...

...[T]he 500 students and faculty in the auditorium that day in 1964 ... sat in absolute silence throughout the talk. When the question period came ... everyone rose and exited, again in silence. So Rockwell began to curse us all. Still no one reacted. Eventually he gave up and left.

The author, AAUP President Cary Nelson, goes on to say "What I learned in 1964 was to value the power of silent, nonviolent witness." Then Nelson gets to his main point: the reaction of various colleges to speaking engagements by Ward Churchill and William Ayers, who, Nelson would have us believe, are really quite nice people once you get to know them. They certainly are not without their controversies. But let's assume they simply follow Oscar Wilde's dictum: "The only thing worse than being talked about is not being talked about."

Nelson recounts several situations where speeches by Ayers, Churchill, and Richard Dawkins have been cancelled due to threat of financial losses or physical violence.

So there you have it. Controversial "right-wing" speakers are afforded an oportunity to air their views on campus, while speakers from the left are driven out by threats and intimidation.

But things are not so neat and tidy. There is a reason that Nelson pulls his initial example from 1964. Things aren't like that anymore, as Tom Tancredo, an opponent of "illegal" immigration, discovered at a recent speech at UNC. Tancredo's speech on immigtration was disrupted by protestors, and ultimately cancelled when protestors broke a window of the room where Tancredo was speaking. You can watch the video here. Now suppose that Tancredo is a real monster, not a "false monster" like Ayers or Churchill. Why is he not accorded the same opportunity as a Nazi?

What does the AAUP have to say about Tancredo's experience at UNC?

Search: tancredo
Found: Nothing!

Violence is not speach. And once violence is injected into politics, it is very difficult to remove.

21-270: Exam #3 Review

The Exam #3 Review Page has been posted.

UPDATE: I've added answers to the review problems.

UPDATE: I've posted solutions to HW#9 on the blackboard site.

UPDATE: I've added solutions to HW#7 to the blackboard site. It looks like I miscounted the number of assignments, as there apparently was no HW#8, but fear not, I've posted solutions to all the problems assigned from Chapters 4 and 5.

Wall Street Bankers Channel Jerry Maguire

Law of Unintended Consequences.

[via Instapundit]

21-270: Assignment for Week #12

The assignment for Week #12 (HW#9) is now posted.

Say It Ain't So.

Please tell me this is from the Onion. Or Scrappleface. Please.

Settlers of Catan Tournament

The CMU Gaming Club will be hosting a Settler of Catan Tournament on Friday, April 10, 2009 from 9:00pm - 1:00am in the University Center's Kirr Commons.

Settlers of Catan is fun, but personally, I prefer Civilization, or even better, Advanced Civilization

21-270: Assignment for Week #11

I've posted the readings for Week #11, and the problems for HW#8.

UPDATE: I inadvertantly assigned problem 3.8, which you did on a previous homework. I've removed it from the assignment. The other problems are correct.

21-270: Change to Office Hours

There has been a scheduling change to Skyler Dong's office hours. The will now be held from 4:00-5:00 on Tuesday and Thursday.

21-270: Exam #2 Results

UPDATE: I've corrected a variety of errors, as pointed out in class yesterday afternoon.

I'm making your Exam #2 grades available on the Blackboard site. I usually don't do this until I've had a chance to return the exams to you, and give you an idea what those scores mean (e.g. what the grade cutoffs are). Tomorrow is the deadlne for choosing the pass/fail option and for withdrawing without a "W" on your transcript. So here is what I can tell you:

Grade cutoffs for Exam #2 are: A, 65-100; B, 50-64; C, 40-49; D, 20-39.

Let me take a moment to explain how I'll determine your final grades at the end of the semester. I'll determine grade cutoffs for your final average by averaging the grade cutoffs for each assignment. For example, to determine the A cutoff, I will compute

.15*[85]+.6*[80 + 65 + (A cutoff for Exam #3)]+.25*[A cutoff for Final].

I'll make a similar computation for the other grades. Currently, those cutoffs are:

Grade cutoffs for Cumulative Average: A, 75.00-100; B, 61.00-74.99; C, 51.00-60.99; D, 34.00-50.99.

The current cutoffs were computed in the following way:

( .15*[Homework Cutoff] + .6*[(Exam #1 Cutoff) + (Exam #2 Cutoff)] )/(.75)

where, for instance, to compute the B cutoff for the cumulative average, you would use the B cutoffs for the Homework and Exams. You can compare your "Weighted Average" from the Blackboard gradebook to these cutoffs to see where you stand.

Here is a comprehensive set of statistics for the course so far.

21-270 Exam #2 Makeup

If you have a conflict with the original exam time, you can
take the scheduled makeup exam: Wednesday morning, 7:30-8:20AM, in WEH 5403.

Please do send me an email to let me know if you will be taking the makeup exam. If you have already written to me, please do not do so again. I'll try to confirm receipt of your message, but don't worry if you don't hear from me.

On Wednesday morning, I will need to know the nature of your conflict. It must be a legitimate conflict (university scheduled event, etc.).

21-270: Exam #2 Review Problems

I've added answer to (most of) the review problems on the Exam #2 Review Page.

21-270: Exam #2 Review

I've posted the formula sheet for Exam #2 on the Exam #2 Review Page. (I made a couple updates to the formula sheet at 3:48pm on Sunday)

xkcd

This one is pretty good, too.

xkcd

As we get ready to begin our discussion of finite probability spaces in 21-270, I think it is apporpriate to think critically about correlation vs. causation.

21-270: Assignment for Week #9

I've posted the reading for Week #9, including the assingment for HW#7

21-270: Assignment for Week #7

I've posted this week's reading assignment, and the problems for Homework #6. It looks long, but I don't think it is as bad as it seems.

Interest Rate Swaps

Interest rate swaps are in the news. Does it count as news if it happened three years ago? I'm not sure, but anyway, there it is.

21-270: Solutions for HW#3

I've posted solutions for HW#3 on the Blackboard site, under "Course Documents."

21-270: Assignment for Week #6

The reading and homework assignment for Week #6 has bee posted

Summer Opportunites in Math Finance

The UT Austin math-finance group is organizing a summer school in math
finance this summer for advanced undergraduates and 1st-year graduate
students. The school will take place in Austin, June 19 - August 8,
2009. Everybody is welcome, space permitting, but we can only provide
financial support for a limited number of students who are US citizens
or permanent residents. More information can be found on the school
website

http://www.ma.utexas.edu/rtg/school2009

21-270: Homework for Week #5

The reading and homework for Week #5 have been posted. You can follow the link from the Schedule page.

21-270: Exam #1 Statistics

Here is the sheet of statistics that I showed in class on Wednesday. It also shows statistics for homework and cumulative averages, including the current grade cutoffs for cumulative averages.

Computational Finance Information Session

On Monday, February 16, there will be an information session for students interested in the computational finance major and minor. The session starts at 7:00 in DH 2315. This letter
gives some more information about the program. At the information session, we will discuss the program, the application process for the major and answer any questions you may have.

21-270: Homework for Week #4

The reading and homework assignments for Week #4 have been posted. The written assignment is due this Wednesday.

21-270: Exam #1 Update

I've added to the review page numerical answers for the practice problems. I've also added a link to the formula sheet you will be given with the exam. There is not that much on it, but there it is.

UPDATE: I've added the solutions to HW#2 to the blackboard site. You can find them under "Course Documents".

BSCF Major Applications

The Spring 2009 deadline for applications to the BSCF major is
Monday, March 2. That is the Monday before the Mid-Semester Break.
Electronic applications may be sent to bscf@andrew.cmu.edu, or hard copies can
be put in Dr. Handron's mailbox in the Mathematical Sciences department office,
WEH 6113.

21-270: Exam #1

I've posted a review page for Exam #1. It shows some general information about the exam. You will probably want to take a look at the suggested review problems. Later in the week, I'll post answers to the problems, so you can tell if your solutions are correct.

21-270: Homework #1 Solutions

Solutions to Homework #1 are now available in the Course Documents section of the Blackboard site.

Economic Stimulus

What did John Maynard Keynes have to say?

21-270: Chapter 2

Chapter 2 of the course notes is now available on the Blackboard site.

A Cause for Concern?

These charts might be a reason to think twice before allowing the Fed to double down.

21-270 Homework for Week #2

The homework for week #2 was posted on Thursday. I forgot to post a note here, though. I've posted a new version that makes one minor correction. In problem 6 part (c), I mistaken typed r^$₁, when I should have typed r^p₁.

If you have questions of comments about the homework, you can post a comment to this thread.

21-270: Historical Revision

One of your classmates wrote to me to correct a historical inaccuracy in a comment I made on Friday. He writes

Today in lecture you mentioned that Robert Merton passed away before
receiving the Nobel prize for the work done in deriving the
Black-Scholes-Merton formula. I read a couple books over the summer,
one particularly about Fischer Black, and I believe from my readings
that Black was the one who passed away before receiving the Nobel
prize. I think that the reason Black-Scholes-Merton formula is more
prevalently known as just the Black-Scholes formula is because Black
and Scholes collaborated together on deriving the formula, using a
method known as the Capital Asset Pricing Model (CAPM). Merton, on
the other hand, believed CAPM to be faulty, but verified the
Black-Scholes formula later on with his own methods. Anyway, I just
thought that was an interesting bit of information.

21-270: Market Efficiency

Market efficiency is not going to be something we give much thought to in 21-270, but if you are interested, this is an interesting data point.

21-270: Things to Keep in Mind #3

Many people have asked about purchasing fractional amount of securities. From page 18 of the notes:

Unless stated otherwise, we assume that all securities can be purchased and sold
short in any amounts we choose (including fractional numbers of shares of stock) and
we ignore transaction costs, including the bid-ask spread. Moreover, we assume the
interest rate for investing is the same as the rate for borrowing (although we do allow
the interest rate to depend on the duration of the loan or investment). We shall
also assume that the price per share of a security is independent of the number of
shares involved in the transaction. We shall also ignore the effects of taxation. A
discussion of the validity of these assumptions from a practical point of view is given
in Appendix II. For assets other than securities, we shall always indicate whether or
not short sales are allowed.

This is as good of a time as any for me to remark that I believe that the lecture notes for 21-270 are by far the best written source for the material covered in this course. If you haven't already, you should begin reading the notes as we go through the corresponding material in class.

Dr. Kramkov is one of the few individuals in the world in possession of extraordinarily deep knowledge of both the theoretical and practical aspects of financial mathematics. He is one of the fields leading researchers and has been responsible for some of its biggest advancements in the last 20 years. Moreover, he served as head of research of one of the world's largest banks. As if that were not enough, he and Dr. Hrusa have the amazing gift of being able to present ideas in a friendly, but thourough manner.

Although the lectures are self-contained, you will not get the most out of this class if you ignore the excellent text.

The way it used to be

A view of the offices of Merrill Lynch, Fenner and Smith from November 1958. From the photographic archives of Life Magazine.

21-270: Calculators

I made this announcement in class on Friday, but I wanted to repeat it here.


Initially, I told you that only one model of calculator would be allowed, the TI-30X IIS. After some questions from students, I've decided to allow a couple other models as well: the TI-30X IIB and the TI-30Xa.


You should note that the TI30Xa does not have the two line display, and doesn't allow you to go back and edit calculations you have entered. This was the model required for last year's class, and students found it difficult to deal with.

21-270: Things to Keep in Mind #2

Another common early mistake is to take the future value of a portfolio by netting out the initial cost. When we speak of value, we do not include any concept of profit; the value of a portfolio at any given time, t, is simply the sum of the time t values (prices) of the component securities. Note that short positions have values opposite in sign to their corresponding long position.

For example, suppose there exists a $4 European call and a $1.50 European put, both struck at $50 on a single share of the same stock initially trading at $50 and both expiring in one year. If we create a portfolio consisting in one call and a short position in two puts, the initial value of this portfolio is the sum of the values of the calls and the shorted puts: $4 + 2 x (-$1.50) = $1.

If, after a year, the stock goes up in price to $60 the puts are worthless and the call is worth $10 (why?). Many students will then calculate the time 1 value of the portfolio to be $10 + 2 x (- $0) - $1 = $9. This is incorrect; the value of the portfolio is simply $10 = $10 + 2 x (- $0).

Likewise. if the stock falls to $45, the portfolio will have value -$10, not -$11.


In general, absolute profit is not as useful of a number to think about as you might first believe. For one thing, it neglects the time value of money. Similarly, it's difficult to analyse without knowledge of alternative investments or a quantification of its associated risk.

21-270: Things to Keep in Mind #1

An early common mistake made by people learning mathematical finance is to think that the value of an option can always be found by using it's pricing formula at expiration.

For example, consider a European call option, currently selling for $8, struck at $50 with expiration T = 1 on a stock with current price S₀ = $60. As you know, if the stock price has fallen to S₁ = $55 in one year, we can calculate the terminal value of the call as:
C₁ = C₁(S₁) = (S₁ - K)⁺ = ($55 - $50)⁺ = $5
When asked for the time 0 value of the call, C₀, many students will argue that:
C₀ = C₀(S₀) = (S₀ - K)⁺ = ($60 - $50)⁺ = $10
This is incorrect. By the value of an option, we simply mean its price; in this example, C₀ = $8. For our purposes price and value are synonymous; we leave any distinction in the meaning of these terms to the Economists.

Valuing (or pricing) the call between time 0 and time T is a substantially more difficult problem and is one of the aims of modern mathematical finance. These valuations are covered in 21-370 and 21-420.

21-270: Homework for Week #1

The Homework Assignment for Week #1 has been posted. You can follow the link from the Schedule page, or from the Blackboard site.

If you have questions or concerns about the assignment, you should feel free to use the comments section of this post.

Extra Carrier Processing

I went to track a package from Amazon yesterday, and this is what I found:

January 13, 2009 12:32:00 AM PITTSBURGH PA US Arrival Scan
January 12, 2009 08:26:00 PM PITTSBURGH PA US Arrival Scan
January 12, 2009 05:09:00 PM PITTSBURGH PA US Possible delay in delivery due to extra carrier processing
January 12, 2009 05:30:00 AM PITTSBURGH PA US Arrival Scan
January 12, 2009 01:54:00 AM PITTSBURGH PA US Arrival Scan
January 10, 2009 02:32:00 AM PITTSBURGH PA US Arrival Scan

Extra carrier processing? Gee. Do you think?

21-270: Chapter 1

Chapter 1 of the textbook is now available in the "Course Documents" section of the Blackboard site.

21-270: Bermuda Options

Yesterday in class someone asked about "Bermuda options," This occurred during the discussion of the difference between European options, which can be exercised only on the expiration date, and American options, which may be exercised at any time before expiration. Bermuda lies between America and Europe, and so, too, with Bermuda options, which may be exercised at certain fixed times before expiration (on the first of each month, for instance).

Bermuda options are generally not traded on exchanges. They are traded in the Over-the-Counter Market, wherein buyers and sellers can negotiate any terms of the contracts (such as which days the option can be exercised).

Other categories of over the counter options include Asian options, digital options (also called binary options or all-or-nothing options) and lookback options.

Welcome to 21-270 Introduction to Mathematical Finance

Welcome to 21-270 Introduction to Mathematical Finance. I'll use this blot to post news and information about the course. The course website has more information about the course: contact information for myself and staff, scheduling information, and general information about class procedures.

There is a blackboard site for the course. I'll use the blackboard site to record grades, and to distribute the class notes and solutions to the homework and exams. Items posted on the blackboard site are there specifically because they are passwork protected, and you do not have permission to repost them elsewhere.

During exams, you will permitted to use one model of calculator only: The Texas Instruments TI-30x IIS. It is a relatively inexpensive, non-programmable, scientific calculator.

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