### Swarthmore College, Spring 2007 Math 53 - Topics in Analysis (Financial Mathematics)

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The FINAL EXAM is now available in an envelope outside my door. It is a take-home exam with no time limit, so there is no reason not to pick it up right away. Warning: I have tried to include some seriously hard parts to challenge you.

Here are two files that might be helpful on the exam: the latest version of BlackScholes2.xls and a normal table, ztable.doc.

4/23/07 -
(1) CORRECTIONS TO EXAM: The reference to "\$100" in problem 1 should be "\$120." The last "+" sign in the main equation in problem 5a should be a minus sign. Both corrections are made in this version of the exam: 53exam2.doc (WORD with MathType, 7 pages).

(2) Here is the long-promised set of notes on deriving the Black-Scholes equation in the risk-neutral case. One section is relevant to problem 1 of the exam. (It might help or might not, but it is relevant). (Word with MathType, 8 pages).

(3) Also, here is a web version of the notes on the two-branch model. You have a paper version, but now it's on the web.

(4) Finally, Here is the spreadsheet used in class last Friday. We used it to find the best mix of XOM, BP, JBLU, and LUV to offset oil future risk, allowing us to buy more futures for the same level of risk.

4/5/07 - Like last week: No problem session at 4-5:30, but yes office hours including 2:30-4 and 5:30-6:30.

4/4/07 - Link to homework 7:
53homework7.doc - 1 page, Word (due 4/9)

4/2/07 - Links to two documents:
MeanVariance2007.doc - Notes on mean-variance optimization, handed out in class 3/30/07. 10-page Word with MathType and graphics. This has been corrected very slightly since the handout version, and might get corrected or revised further.
AppleMicrosoft.xls - Spreadsheet for minimizing the risk in a portfolio of AAPL and MSFT (and maybe GE and IBM). From class 3/30 and 4/2.

3/29/07 - NO PROBLEM SESSION AT 4. I made a scheduling error; I can't be at a problem session from 4 till 5:30 today. I'll be in my office to talk about the problems from before 3 till almost 4, and from 5:30 till 6:30.

3/26/07 - Remember to save the date: April 10, colloquium with Doug Costa from Susquehanna Investment Group.

3/26/07 - Current links to best versions of some workbooks:
BlackScholes2.xls - Black-Scholes formula with delta, vega, and European puts
valuation.xls - The giant grid for evaluating put options and exotic options. (It's set for manual calculation. Press "F9" when it's time to recalculate.)

3/26/07 - Homework 6: 53homework6.doc - Word, no MathType.

3/22/07 - Is there any interest in a problem session today (Thursday)? I'll be in my office at 5:30 (I have a seminar that ends about then). But I won't stay long if nobody is there. I'll also be around most of 11-1 and 2:30-4, which might be better times for questions.

3/20/07 - I'll just keep updating that "transaction" page without changing the date on this page. Here's a link to solutions to problem set 5.

3/12/07 - UPDATED 3/16 4:30pm-->Click here for notes on last Friday's option trades, and how to keep track of the results. Through 11:45 Monday: no change worth noticing. Also, here's an updated version of the Black-Sholes worksheet. I'm not sure I saved the version we were actually looking at in class, so I have revised this one to show the actual purchase price of the IBM shares, 92.95. I have also adjusted the volatility (to 0.1681) to reconcile with the 14.20 midpoint price of the K=90 option. With those adjustments the theoretical price of the K=130 option is \$1.69, which would mean that it really was underpriced at 1.525 (midpoint price) or 1.60 (actual purchase price).

3/1/07 - Here is a link to a new version of the Black-Scholes worksheet, with lines added for "delta" and "vega". (Recall: delta = partial of V wrt S0, vega = partial of V wrt sigma.) There is also a second page in the workbook, which can be used for designing a mix investments that is "delta-neutral" and "vega-neutral". (That is, the portfolio is insensitive, to first order, to changes in the stock price OR the market's opinion of volatility.) We'll do that next week, I think.

Friday we'll finish the utility theorem (easy version and what we can do of the full version). But before we move on to the arbitrage theorem, I want to introduce the two-branch model that plays such a big role in dealing with non-risk-neutral option pricing. So we'll do that Friday, too.

2/23/07 - Two new documents under "course documents."
(1) Homework 5 is posted. (Sorry, this and other links were bad, now fixed.) (2) A big Word document (35 pages, shaped like a slide show) from a short course I gave in 2005: PartA2005a.doc (Word, 35 pages with MathType and graphics). This contains, among other things, a derivation of the Black-Scholes formula under the assumption of risk neutrality, close to the derivation we had in class.

There is also a page on the relationship between R(t) (daily returns measured logarithmically) and A(t) (same, measured arithmetically). Looked at individually, they are almost indistinguishable. But if you look at their means, which are both close to zero, you find
mean(A(t)) = mean(R(t)) + (1/2) variance(R(t)).
This is that "one-half-sigma-squared" term that keeps popping up.

2/19/07 - Also, the Black-Scholes Excel spreadsheet from class 2/16 is finally up under course documents and here. The first column is set up with the inputs for the IBM option we looked at. The actual price of the option is about 12.90. We'll try to reconcile the difference today.

2/17/07 - Homework 4 wasn't on the website, but it is now, under course documents.

2/14/07 - Not a snow day today; here I am already. But check out this link.

2/7/07 - Today we'll introduce a "stochastic process" called "geometric Brownian motion" (GBM). It's the generic name for the standard model of share prices. Notes are listed under "course documents" below.

2/6/07 - It's a little late, but I have revised the web version of HW 3 to clarify some points.
(1) The P(t) functions in problems 1 and 2 are different. These problems are independent of each other.
(2) For problem 4 (translating to flat dollars) use the P(t) function from Problem 2. (In particular, P(1/2 year) = 0.99.)
(3) Also, here's a more complete definition of "flat dollars:"
“A flat dollar delivered at time t is the same as (1/P(t)) US dollars delivered at time t. Equivalently: a US dollar delivered at time t is the same as P(t) flat dollars. ”
(4) The P(t) values in the strip-prices-29jan07 spreadsheet are multiplied by 100. So, if the table seems to say that P(t)=98.553, then P(t) is really 0.98553.

2/6/07 - The above "schedule" link works now. The schedule is pretty rough, but it does now reflect the plan to go as far as we can without risk aversion. In particular, we will derive the Black-Scholes option-pricing formula under an assumption of risk neutrality. That way, we can get the calculations over with in a relatively simple context. Then we'll see how the same results can be obtained without the risk-neutrality assumption.

2/4/07 - Homework 3 is posted under "course documents."

(1) Find http://finance.yahoo.com
(2) Enter a symbol. To find the symbol for a company, click on "look up symbol."
(3) At the left side of the page, find the link for "historical prices"
(4) Pick your favorite dates (at least 2001-2006)
(6) Store the resulting ".csv" file or open in Excel
(7) Copy and paste the data (or just dates and "adjusted closing") to a workbook under your control.
For each day, compute R(t)=ln(S(t)/S(t-1)). The easiest way to find the mean and standard deviation for the series of R(t) values is to enter the formulas
= AVERAGE ( B2:B1500 ), and
= STDEV ( B2:B1500 )
where "B2:B1500" is replaced with the range of your R(t) values.

1/31/07 - Solutions to homework 2 are posted under "course documents." So is the four-page handout from class today.

www.treasurydirect.gov/ - The Treasury's site for bond info
www.treasurydirect.gov/instit/marketables/tbonds/tbonds.htm - their site for bonds
www.treasurydirect.gov/instit/marketables/strips/strips.htm - their site for STRIPS

WSJMarkets.com - Wall Street Journal's free data site; lots of good stuff but no STRIPS

200042pap.pdf - a Federal Reserve paper on estimating yield curves from STRIPS (pdf)

www.zionsbank.com/zd_bonds_about-guest.jsp - a commercial site with STRIPS quotes. Click on "Treasuries" and then select STRIPS; exclude inflation-adjusted items and callables.

bonds-29jan07.xls - might be used in class 1/29
strip-prices-29jan07.xls - might be used in class 1/29

1/24/07 - Solutions to homework 1 are posted under "course documents."

1/24/07 - Homework 2 is posted under "course documents." Also, here is a set of notes on interest rates that I'll hand out today (Word with MathType, 7 pages).

1/23/07 - HEY, THERE'S A COOKIE FRIDAY this Friday, 3:30-4:30, in the math/stat living room. You're invited!

1/23/07 - There is a typo in Homework 1. You can probably just ignore it, but here's the correction: In problem 7, the constants are T and n, not T and tau. (Tau is the variable with respect to which the limit is taken.) As long as we're changing the problem, let's change the name of "n" to "x" --- so the problem becomes lim(as tau -> 0 from above) of (1 + x tau)^(T/tau), where T and x are constants. The versions on the web have been corrected.

1/22/07 - Homework 1 is posted under "course documents." So is the questionnaire, but I'll have a few extra paper copies of that on Wednesday.

1/16/07 - A TRADING OPPORTUNITY: If you have any loose foreign exchange lying around --- coins or small bills --- bring it to class on the first day and we'll set up a foreign exchange market. You can get rid of useless change or maybe snap up bargains.

1/16/07 - Here's the text. There's a link, above, to the text's website.

"Mathematics for Finance: An Introduction to Financial Engineering"
Marek Capinski and Tomasz Zastawniak
1-85233-330-8 (paperback)
I have just alerted the bookstore, so it won't be there for a few days. But you can get it from Springer (www.springer.com for the company, or this link for the book) or from Amazon for about \$30. Amazon has some kind of system for upgrading to a searchable electronic version; if you know anything about that, please let me know.

I won't follow the text very closely, but it will give you a fighting chance to see someone else's take on the subject. I'll at least use some of the text's notation. The book has 11 chapters, and we will cover most of what is in chapters 1-5, 7-8, and 10, some of it more deeply than in the text. For example, we'll pay more attention to the "geometric Brownian motion" process in continuous time, and spend some time on the statistics of actual stock prices.

11/22/06 - Here's a VERY preliminary essay on the probability prerequisite.
53ProbabilityPrerequisite.doc (4-page Word document with MathType)
53ProbabilityPrerequisite.htm -- web version (symbols may not come through)
A better reference is this:
Grinstead and Snell, Introduction to Probability, 2nd ed., Chapters 1-6 except sections 3.3 and 4.3.

11/15/06 - Here's a possible outline of topics:
TopicsInAnalysis.htm -- html version
TopicsInAnalysis.doc -- 4-page Word version
This is pretty rough, but it is a good guide to what we'll do during the semester. I hope to post a statement soon about prerequisites (how much probability?) and the mathematical nature of the course. (Will we prove theorems? Yes, but we'll take a more computational approach than we would expect in, say, Math 63.)

53homework7.doc - 1 page, Word (due 4/9)

MeanVariance2007.doc - Notes on mean-variance optimization, handed out in class 3/30/07. 10-page Word with MathType and graphics.
AppleMicrosoft.xls - Spreadsheet for minimizing the ris k in a portfolio of AAPL and MSFT (and maybe GE and IBM). From class 3/30 and 4/2.

53solutions5.doc -- Solutions to homework 5, 3/20/07. 3-page Word with MathType.
transaction.html -- running report on option trades. Updated from time to time.
PartA2005a.doc -- Notes from an earlier MAA short course on this subject. Word, 35 pages.

53homework5.doc -- Homework 5, 2/23/07, due 2/28/07 (2-page Word with MathType)

Black-Scholes.xls -- A one-page Excel worksheet for calculating the value of an option using the Black-Scholes formula. From class 2/16/07.
Black-Scholes1.xls -- Improved version of same, shown in class 2/28, with a second page for designing delta-and-vega-neutral portfolios.

53homework4.doc -- Homework 4, 2/12/07, due 2/19/07 (Word with MathType)

GBM1.doc -- Notes for class 2/7/07; 4-page Word with graphics and MathType

53homework3.doc -- Homework 3, 2/1/07, due 2/7/07

Interest5.doc -- more notes on interest: yield on bonds, pure risk-free yield curve, forward interest rates. 1/31/07, Word with MathType, 4 pages.
Interest4.doc -- rough notes on interest, bonds, and yield curves, 1/24/07, Word with MathType, 7 pages.

53solutions2.doc -- Solutions to homework 2, 1/31/07, 3-page Word with MathType.
53homework2.doc -- Homework 2, due 1/31/07, 2-page Word document with MathType, 1/24/07
53homework2.htm -- Same, html format

53solutions1.doc -- Solutions to homework 1, 1/24/07, 1-page Word with MathType.
53homework1.doc -- Homework 1, due 1/24/07, 1-page Word document with MathType, 1/22/07
53homework1.htm -- Same, html format
53questionnaire.doc -- questionnaire, 1-page Word document, 1/22/07
exchange-rates-1-21-07.xls -- Excel workbook from class 1/22/07

53ProbabilityPrerequisite.doc -- about the probability prerequisite (4-page Word document with MathType)
53ProbabilityPrerequisite.htm -- web version (symbols may not come through)

TopicsInAnalysis.htm -- preliminary outline of topics, html version
TopicsInAnalysis.doc -- 4-page Word version

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