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!
Mike's Math Page 