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Archive for March, 2010

Last February, I wrote about the current financial crisis and the role that mathematical trading strategies played in the meltdown.

I recently finished reading The Quants by Wall Street Journal reporter Scott Patterson.  In The Quants Patterson tells the story of computer trading strategies at the big NY investment banks and various hedge funds.  He covers some of the history of quantitative analysis of financial markets as well as the meltdown that started in August of 2007.

One paragraph in particular caught my eye as I was reading:

The market moves PDT and other quant funds started to see early that week defied logic.  The fine-tuned models, the bell curves and random walks, the calibrated correlations – all the math and science that had propelled the quants to the pinnacle of Wall Street – couldn’t capture what was happening.  It was utter chaos driven by pure human fear, the kind that can’t be captured in a computer model or complex algorithm.  The wild fat-tailed moves discovered by Benoit Mandelbrot in the 1950s seemed to be happening on an hourly basis.  Nothing like it had ever been seen before.  This wasn’t supposed to happen. (Italics in original)

Things that the mathematical trading models had predicted to be so unlikely as to happen only once in 10,000 years happened three days in a row in August 2007.  This is not just like flipping a coin and getting 10 heads in a row, it’s like flipping a coin and having it land on edge 10 times in a row.

[By the way, if your model predicts that something will happen once in 10,000 years and then it happens three days in row – NEWSFLASH – there is something deeply flawed in your model!]

Part of the problem was laid out by Nassim Nicholas Taleb, author of the book The Black Swan.  The mathematics used in the world of physics and other hard sciences uses standard bell curves.  If you measure the height of 1,000 people off the street, even if you include a few NBA players, the average won’t change all that much.

The problem in finance is that the scale of the financial world is radically different from the scale of the physical world.  If, instead of measuring the height of 1,000 people, you are measuring the financial net worth of 1,000 people, having someone like Bill Gates in your sample of 1,000 can have extreme effects on the average.

This plays out in the financial world in the effect that very large pools of money inevitably have on markets as they move in and out of various trading positions.  Standard statistical models that were developed to deal with the natural world don’t account for the effect these outsized moves have on prices and markets.

Another idea that I found interesting in The Quants was that, as quantitative trading strategies caught on through the 90s and 00s, more and more people began using similar strategies, which made these strategies less profitable.

There are only so many trades to go around, so if, instead of $1 billion chasing these slight pricing inefficiencies, you suddenly have $100 billion and more chasing these same slight inefficiencies, there is less and less profit in each trade.

The solution – LEVERAGE, LEVERAGE, LEVERAGE.  In other words, once the profits on these strategies started becoming smaller and smaller (because they were effectively being split between more and more traders using similar strategies), they needed to use bigger and bigger trades to get the same amount of profit out.

Many of the hedge funds and investment banks were leveraged 10, 20 or even 30 to 1.  For example, in The Quants, Patterson describes the hedge fund Citadel Group as having $140 billion in assets on only $15 billion in actual capital.  This is about a 9 to 1 leverage ratio.

What makes this so important is that if these trades go bad on the hedge funds, they stand to lose A LOT more money than they can conceivably pay back, especially if what they’re invested in turns out to be worthless.

So, when things turned bad in August of 2007, all of a sudden, all the hedge funds that were making these trades with vast amounts of borrowed money were scrambling for the exits all at once.  They were all highly leveraged and they were all in almost the same trading strategies which meant they were all trying to get out of the same doorway at the same time.

In a 60 minutes interview with Steve Kroft last year, Frank Partnoy pointed out that “You can’t model human behavior with math.”

Another way of saying this is that finance and economics are social sciences – not hard sciences.

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I posted about the Complex Numbers last summer, and also wrote about using Newton’s Method to find complex roots.

The New York Times math blog by Professor Steven Strogatz of Cornell University has an excellent post on complex numbers and Newton’s Method with some great graphics as well.

It ends with a wealth of sources on complex numbers, fractals and Newton’s Method and the interrelations among them.

The other posts from his blog are also well worth reading.

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A presentation on M. King Hubbert‘s adaptation of Pierre Verhulst‘s logistic function to model oil production.

The proof that a paraboloid of revolution reflects parallel waves to a single point.

How to calculate distance between two locations on a sphere given latitude and longitude.

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