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Archive for February, 2009

Because the math is really complicated people assume it must be right.” — Nigel Goldenfeld, whose company sells derivatives software.(from a NYTimes article March 9, 2009)

The cover article in this month’s Wired magazine has to do with the applications of mathematics to finance and investing.  Specifically, it talks about what is called the Gaussian Copula function, which supposedly allowed investment banks, hedge funds, and other high-flying investment professionals a quick, straightforward way to determine the risk in a pool of investments.

The article starts out explaining that the mathematician who came up with the Gaussian Copula function was widely celebrated and that many people even thought that he might win a Nobel Prize.

“A year ago, it was hardly unthinkable that a math wizard like David X. Li might someday earn a Nobel Prize. After all, financial economists—even Wall Street quants—have received the Nobel in economics before…”

What the article fails to mention, and I think that this is a particularly egregious omission, is that the last mathematical/economics investment wizards to win the Nobel Prize were the team behind the Black-Scholes equation –

“Robert C. Merton and Myron S. Scholes have, in collaboration with the late Fischer Black, developed a pioneering formula for the valuation of stock options. Their methodology has paved the way for economic valuations in many areas. It has also generated new types of financial instruments and facilitated more efficient risk management in society.”

Great! So the mathematicians have finally figured out how to feed a computer equations and get it to spit out money in return!

Wrong.

From Wikipedia

“Together with Myron Scholes, Merton was among the board of directors of Long-Term Capital Management (LTCM), a hedge fund that failed spectacularly in 1998 after losing US$4.6 billion in less than four months.  The Federal Reserve was so concerned about the potential impact of LTCM’s failure on the financial system that it arranged for a group of 19 banks and other firms to provide sufficient liquidity for the banking system to survive.”

Long-Term Capital Management was like a dress rehearsal for today’s financial meltdown.  I think that it would have served the author well to at least mention that the last widely celebrated financial risk assessment mathematicians were also spectacularly wrong.

Here’s the problem – people want a short-cut.  Math has some great short-cuts.  Think of the quadratic formula in Algebra, or the power rule for differentiation in Calculus.  But, for some things, there is no short-cut.  There is only the grunt work of going in and examining and analyzing enough pieces of what you’re working with so that you have a solid overall understanding of the big picture.

When David X. Li came out with his new short-cut for assessing investment risk…

“The effect on the securitization market was electric. Armed with Li’s formula, Wall Street’s quants saw a new world of possibilities. And the first thing they did was start creating a huge number of brand-new triple-A securities. Using Li’s copula approach meant that ratings agencies like Moody’s—or anybody wanting to model the risk of a tranche—no longer needed to puzzle over the underlying securities. All they needed was that correlation number, and out would come a rating telling them how safe or risky the tranche was.”

Normally, ratings agencies and/or investors have to do a lot of analysis to determine the quality of investments.  This is what is known as DUE DILIGENCE.  This is a very important legal term that essentially means doing your homework and NOT taking short-cuts.

Two mathematicians who have their own ideas about applications of math to investing are Benoit Mandelbrot and Nassim Nicholas Taleb.  I wrote a blog entry about these two back in October.

Mandelbrot is the mathematical father of fractals and fractal analysis.  He published a book a few years ago about the application of the ideas of fractal analysis to the financial markets.

Nassim Nicholas Taleb’s training and education were focused on the application of mathematics and statistics to business and finance.  He wrote the book The Black Swan, about how improbable occurrences  that we are unaware of can have a dramatic impact on what we expect the future to look like.

Both of these mathematicians point out the severe limitations of both the Black-Scholes model and the Gaussian Copula model.  The Wired article discussed Taleb’s take on the role the Gaussian Copula function played in the recent financial meltdown.

Nassim Nicholas Taleb, hedge fund manager and author of The Black Swan, is particularly harsh when it comes to the copula. ‘People got very excited about the Gaussian copula because of its mathematical elegance, but the thing never worked,’ he says. ‘Co-association between securities is not measurable using correlation,’ because past history can never prepare you for that one day when everything goes south. ‘Anything that relies on correlation is charlatanism.’ “

On the other hand, some people have used the Black-Scholes model successfully by understanding its limitations.

Paul Wilmott, who is quoted in the Wired article, has also been critical of attempts to quantify risk.  But, he has used the Black-Scholes model in the past to develop trading strategies.

“A couple of years after leaving academia I became a partner in a volatility arbitrage hedge fund, and this was the start of phase three. In this fund we had to price and risk manage many hundreds of options series in real time. As much as I would have liked to, we just weren’t able to use the ‘better’ models that I’d been working on in phase two. There just wasn’t the time. So we ended up streamlining the complex models, reducing them to their simplest and most practical form. And this meant using good ol’ constant volatility Black-Scholes, but with a few innovations since we were actively looking for arbitrage opportunities. From a pragmatic point of view I developed an approach that used Gaussian models for pricing but worst-case scenarios for risk management of tail risk. And guess what? It worked. Sometimes you really need to work with something that while not perfect is just good enough and is understandable enough that you don’t do more harm than good. And that’s Black-Scholes.”

The problem is not necessarily the mathematics but that people often take mathematics (and science) as some kind of all-seeing oracle that will make decisions for them.  This blinds them to what is actually happening right before their eyes and they assume that they don’t have to do the (intellectual) heavy lifting, because the equations and computers are doing that for them.

Remember the fundamental mantra of computer programming –Garbage In, Garbage Out

The Wired article says that

“Bankers should have noted that very small changes in their underlying assumptions could result in very large changes in the correlation number. They also should have noticed that the results they were seeing were much less volatile than they should have been—which implied that the risk was being moved elsewhere. Where had the risk gone?

They didn’t know, or didn’t ask. One reason was that the outputs came from “black box” computer models and were hard to subject to a commonsense smell test. Another was that the quants, who should have been more aware of the copula’s weaknesses, weren’t the ones making the big asset-allocation decisions. Their managers, who made the actual calls, lacked the math skills to understand what the models were doing or how they worked. They could, however, understand something as simple as a single correlation number. That was the problem.”

And with the money rolling in, nobody wants to ask questions.

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Georg Cantor was an important mathematician who lived during the late 19th century.  He devised a method for comparing different sizes of infinite quantities and demonstrated that there were differnet sizes of infinite sets.

The foundation of Cantor’s ideas is the concept of a one-to-one correspondence.  This is a concept which is deeply rooted in the human psyche, as is shown by the fact that the word digit – meaning number – also means finger.  This one-to-one correspondence between enumerating objects by comparing them to the fingers and/or toes of the human body is the presumed origin of our base ten number system.

What Cantor did was to compare infinite sets by establishing a one-to-one correspondence bewteen the elements of the two sets.  If it was possible to establish a one-to-one correspondence, then Cantor concluded that the sets were the same size.  In 1873, Cantor published a paper showing that the rational numbers (which includes all fractions and whole numbers) is actually the same size as the counting numbers (1, 2, 3, …and so on).  Any set that is the same size as the counting numbers is called “countable.”

Another example of this is the set of even numbers.  The set of even numbers is the same size as the set of counting numbers.  It might seem as though there would be fewer even numbers, because the set of counting numbers includes all the odds, but the set of even numbers doesn’t!

But, you can create a one-to-one correspondence between the coutning numbers and the even numbers.

1 (in the counting numbers) matches up with 2 (in the even numbers)

2 (in the counting numbers) matches up with 4 (in the even numbers)

3 matches up with 6

4 matches up with 8 and so on.  Any counting number N will match up with 2N in the set of even numbers.

For every counting number, there is a corresponding even number – therefore the sets are the same size.  Cantor called this size – aleph null, the smallest infinite size.  Cantor showed that the Real Numbers, represented by the number line and including all rational and irrational numbers, are NOT countable – in fact, there are more Real Numbers than Counting Numbers.  Cantor’s famous Diagonal Proof is actually fairly simplistic.

Most mathematicians of the late 19th century hated Cantor’s ideas, and he didn’t receive much support intellectually while he was struggling with his work.  Cantor often became depressed toward the end of his life and spent time in and out of sanitariums, eventually dying in 1918.  By this time, however, many younger mathematicians looked up to Cantor and relied heavily on his work and ideas.

Today Cantor is regarded as one of the most important mathematicians to ever work.  His ideas regarding infinity are deeply philosophical and have had a profound impact on mathematics, science and philosophy.

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