Being a weblog devoted to a variety of topics. Including Mathematics. And Mathematical Finance. Sometimes with homework.

Wednesday, January 28, 2009

Economic Stimulus

What did John Maynard Keynes have to say?

Tuesday, January 27, 2009

21-270: Chapter 2

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

Sunday, January 25, 2009

A Cause for Concern?

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

Saturday, January 24, 2009

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 rp1.

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

Wednesday, January 21, 2009

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.

Monday, January 19, 2009

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.

Friday, January 16, 2009

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.

Thursday, January 15, 2009

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.

Wednesday, January 14, 2009

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?

Tuesday, January 13, 2009

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.

Monday, January 12, 2009

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.

Friday, January 9, 2009


Windows 7

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