My math history and Black-Scholes

I had the benefit of a great math teacher in high school – Mr. Waterman – as well as a great principal – Dr. Moller – who gave his department heads an enormous amount of flexibility as to how they ran their departments.  Because of this nice bit of luck I was able to learn a great deal of math in high school in Omaha.  Even topping off my senior year with a course in differential equations.

My senior year of college also ended with an interesting experience in differential equations.  In the spring of my junior year a team of scientists (who would win the Nobel prize in 2006) found some interesting structure in the cosmic microwave background radiation.  My undergraduate physics thesis looked at the possibility that a rotating universe might help explain some of this structure.   While I was finishing up my research I ran across a differential equation that was well outside of my ability to solve.   It was not beyond the ability of my thesis advisor, Ed Bertschinger, though, and his ability to seemingly wave his hands and draw a solution on his chalkboard amazed me.  After a little more work on my end,  computer solutions of the equation confirmed his hand waving was exactly right.  It was a great lesson in really understanding solutions of equations.

In the third year of my PhD program I began to get interested in finance and  ended up taking a few courses at the business school.  One of the professors asked me to give a talk on a paper by  Zvi Bodie of BU – “On the Risks of Stocks in the Long Run.”    In the paper Bodie asks and answers a pretty simple question that gives an enormous amount of insight in to the Black-Scholes option pricing formula.    The question is essentially this:  if you invest $100 in stocks, how much would a put option cost that would guarantee that your original $100 investment will at least return the risk free rate to you.  One reason this simple question is so brilliant is that all of the complicated structure of the Black-Scholes equation collapses and the solution is incredibly simple.    The biggest surprise  to me was that the solution is independent of the risk free rate – meaning that you could study this paper using the simple case that interest rates were always zero.

Bodie’s paper gained a lot of notoriety because of a second surprising result – the cost of this insurance increases over time.  The cost as a function of time given in his paper is as follows:

Time in years            Cost of put as a percent of initial investment assuming 20% volatility

1                                7.98

5                               17.72

10                             24.84

20                            34.54

30                            41.63

50                            52.08

75                            61.35

100                          68.27

200                          84.27

So, not only does the cost of the insurance increase over time, the cost actually goes to 100% of the initial investment as the time periods gets large.

From a purely mathematical perspective, this result stands.  If you believe Black-Scholes, you have to believe this result of Bodie’s paper.  The paper’s conclusion means that for very long time horizons the company writing the put is getting paid a price that effectively assumes stock prices are going to zero with probability 1.  The reason I make this claim is because for long time periods you are getting paid nearly 100% of the present value of your maximum payout.  More simply, assume interest rates are zero (since the results are independent of interest rates) and the investor wants to protect a $100 investment, for long time horizons they are paying you nearly $100 to purchase the protection and the most you can ever pay back to them is $100.

This growth of the cost of the put over time really interested me, and I began to dig a little deeper into the paper and the Black-Scholes equation in general.  With a little work I found that I was able to replicate the results in the paper by simply assuming that the stock market declined by an expected 2% per year (again using zero interest rates for simplicity and using the same 20% volatility that Bodie used).  That meant that seemingly odd result in the paper – that over long periods of time the cost of insurance is nearly equal to your maximum payout – actually isn’t surprising at all.  The results of the calculations are equivalent to a case where the market does actually go to zero over time.

Back in graduate school this was all a purely a theoretical exercise.  I was not involved in the financial markets in any way.  Even if I had been, the standard way of trading puts involves posting collateral.  As we saw in the financial markets in 2008, collateral posting can drastically change the economics of any long-dated financial contract.  Chapter 7 of “When Genius Failed” discusses the hedge fund Long Term Capital Management’s experience in the long-dated put market, as a second example.

Ultimately, though, I put the idea in the back of my mind as an interesting mathematical curiosity.  If the opportunity to write longer-dated puts were to arise, I’d at least have some knowledge about it.

The main point, though, is that the mathematical training that I’d been lucky to have going back to high school had come into play on this project.   I keep this example in the front of my mind when I’m teaching my kids.    I try to expose them to a broad spectrum of math and problem solving just like Mr. Waterman did for me and all of his students.  I also try to give them some exposure to problems that are a little over their head so then can see (what for them will be) advanced problem solving techniques like I saw from Professor Bertschinger.  You never know when and where these lessons will come into play.

Finally, since I mentioned “When Genius Failed” above, I’d be remiss if I didn’t mention one of the best quotes on Black-Scholes in the book – “The MIT types always want to short volatility” from Citigroup’s Andrew Hall.  In English, he’s saying that the quantatative types always want to sell insurance on the stock market. So take this point with a grain of salt!

James Tanton’s Infinite series video and paper tearing

Yesterday James Tanton posted a fun video about an infinite series on Twitter:

In this video he presents a fun way to understand an infinite geometric series.  I’d not seen this approach before and thought it would be great to run through with the boys.  It is always nice to see a new (and relatively straightforward!) way to introduce an advanced mathematical concept to kids.  Here’s what we did:

Definitely a fun exercise to start the day.

For another fun math and paper exercise for kids try paper folding.  Naturally, James Tanton has set the bar high here, too:

Learning about that record attempt actually inspired our first Family Math video:

For another fun post about infinite series, check out Patrick Honner’s blog here:

Mr. Honner’s blog has quite a bit about paper folding exercises, too, of course.  A long list of fun activities is here, including one based on a really cool problem from the 2011 AIME!!

Quadratic and Cubic equations

About two weeks ago my son asked me a fun question about cube roots.  We’d spent a lot of time studying quadratic equations over the last couple of months, but for some reason he was looking at cube roots rather than square roots this morning (I don’t remember why).  The problem he was working on required him to find the cube root of 343.  He asked me why there was only one solution here but square roots had two solutions.

I’d been looking for something to talk about over Christmas break anyway, and this random little question definitely was ready made for a week of talks.  We spent a total of seven days discussing quadratics and cubics and had a lot of fun.

(1) The first day we spent some time playing around on Mathematica looking at the difference between graphs of quadratic equations and graphs of cubic equations  (with real coefficients).  After doing that, we talked about what we saw:

(2) The next day we went back to look at a little algebra.  The algebra book we are studying (Art of Problem Solving’s “Algebra”) had sections on differences of squares and  differences of cubes.  We’d spent so much time away from cubes, though, that I didn’t expect that my son remembered much about differences of cubes.  We revisited that topic in this video and show how we can use the quadratic formula to help us find all of the roots of an equation that is a difference of cubes:

(3) In the last video we found the three roots of the equation x^3 = 1.  In this video I tried to show a little bit of geometry behind these numbers.  I wanted to do talk through some of the geometry before doing a bit more arithmetic with the imaginary numbers.   We also ended up spending a lot of time in this video talking about the equation x^4 = 1.

(4) Here we dig into the geometry of the solutions of x^3 = 1 a little more.  In particular, we check that the three solutions all lie on a circle of radius 1 in the imaginary plane.

(5) Next up, we finally get around to checking whether or not the three solutions we found to x^3 = 1 are actually solutions.  So, this video is a little bit of arithmetic on complex numbers.

(6) Having now gone through quite a bit of algebra, geometry, and arithmetic, I wanted to spend a little time on history.  The point was to give a little context to the question my son asked and explain that polynomials with degree “n” have n solutions.  The fact that we find two solutions for quadratics and three solutions for cubics was just a special case of a more general fact.

(7) I didn’t feel like we should end on the fundamental theorem of algebra, so we did one more little talk.  In this one we talk through the history of finding solutions to polynomials.  We start with the quadratic equation and end with Galois.

All in all, this was a fun way to spend time on Christmas break.  Happy that my son asked me about cube roots last week!

Some fun with Archimedes and Pi

Last night I was flipping through one of my favorite books – 100 Great Problems of Elementary Mathematis:

Book pic


I came across the section describing Archimedes’s method of calculating pi and thought it would make a fun morning activity with the boys.  Some of the details of the geometry are a little over their heads, but I didn’t want to get too caught up in the details anyway.  Made for a fun morning.

The first part was just kind of silly – finding a way to introduce pi:

From here we moved on to Archimedes’s method.  We drew a hexagon inscribed in a circle and showed how you could use that hexagon to get the simple estimate that pi = 3:

Playing around with the hexagon inside a circle turned out to be fairly straightforward.  Next we moved on to the slightly more difficult problem – discussing a hexagon circumscribed about the circle.  We studied the picture for a while, used the Pythagorean theorem, and eventually found the perimeter of this hexagon, too.

After this, my instincts led me astray initially.  Luckily, though, I caught myself before moving on, and spent a little time asking the kids to see if they could figure out how to improve our estimate:

The next part is probably the most difficult.  Archimedes figured out a really neat relationship the perimeters of certain polygons and used that relation to get better and better approximations to pi.  The derivation of the relation really uses only the Pythagorean theorem, but I didn’t want to get caught up in the details today:

Finally, we moved to the computer to run through some of the approximations.  I’ve been trying to figure out ways to incorporate a little more computer math, so I was really happy to have the opportunity to do that here.   The last step in our little talk about pi was writing a simple program:

The simple program that we wrote to study the formula is easy to share, you can play with it here if you want:

The kids seemed to really hearing about this way to calculate pi.  It was fun having the chance to walk through all of this with them.