Week 2: Drunken Meanderings, Fat Tails, and the Markets

Hey guys!  I hope you are all having a fun and productive time at your internships.  This week, I thought I would address some of the earliest research in econophysics.  (Buckle in for another unnecessarily long blog post, everyone!)
Econophysics, it turns out, isn't nearly as innovative as it sounds.  As I mentioned last week, the term was coined almost twenty years ago, (in 1995) but the field's central tenants have been around for much longer than that.  In this blog post I'm going to be talking about Louis Bachelier and his Theorie de la Speculation, or in English, Theory of Speculation.  My onsite adviser, Dr. Frieden, suggested that I start with his work, since it has greatly influenced economic and econophysical theory since its rediscovery in the mid 1950s.

Bachelier, originally trained as a physicist, published his Theory in 1900, dealing principally with the movements of stock prices.  (As data, Bachelier analyzed numbers from the Paris Bourse, then France's principal financial exchange, where he worked for several years in order to pay for his schooling.)

Bachelier's work had two parts that are significant to modern econophysics; I'll explain them both below.  First of all, he derived the mathematical expression of Brownian motion.  Second, he found that the motion of prices in a stock market follows a statistical random walk model.

Since neither of those probably meant very much to you, I'll explain.  Brownian motion is the seemingly random motion of particles suspended in a fluid.  (Liquid or gas.)  A common and easily visualized example is the motion of dust molecules in the air.  This gif (credit to Wikipedia) shows a big particle (e.g. dust particle) colliding with a lot of smaller ones (e.g. gas molecule) and producing random, Brownian motion.

(For those of you who worked with Mr. Young in Multivariable Calc this year, I actually talked to him about this, and it turns out that his work in mathematical modeling is similar to modeling Brownian motion; however, he assumes that the particles or objects he is dealing with have some initial velocity that can impact their final positions.)

When Einstein published a paper in 1905 describing Brownian motion, he didn't know that he was five years too late.  Actually, no one did, as Bachelier's work languished in obscurity for decades despite the fact that he invented a lot of probability theory that was years before its time.

While it is interesting to note that a physicist derived a way to understand our natural world by looking at the economy (which perhaps belies an intrinsic connection), Bachelier's work had a more important side effect that lay in the expression of Brownian motion; if given an infinite amount of time, Brownian motion begins to follow a statistical random walk model.

What's a random walk?
Imagine an incredibly drunk man trying to find his hotel room in a long hall.  Since he's stumbling around, blindly intoxicated and completely clueless, he has an equal probability of stepping forward as stepping backwards.  If he steps forward with his first step, he could step either forward or backward with the next.  You might even say that he's walking... randomly.

It turns out that these random walks follow our buddy the normal curve!  This makes intuitive sense; if our drunk friend starts at hotel room number 50, in principle he could end up at room 10 or room 100 by taking a whole lot of drunk steps forward, or a whole lot backwards.  These "extreme" events are just a lot more unlikely because they would require a certain result (stepping only forward or backwards) many times in a row, similar to flipping a coin 50 times and getting 50 heads.

If you'll recall, though, Bachelier's work was not originally about Brownian motion or random walks at all-- it was about stock prices.  Essentially, Bachelier found that the motion of stock prices follows a random walk.  (It was only as he explored the mathematics of random walks that he "accidentally" solved Brownian motion.  Oops.)

This should be fairly intuitive to us also; we would expect a stock price to fluctuate, but not very much.  What Bachelier was arguing was that small fluctuations in price were much more likely than large ones; statistically speaking, a stock jumping from $1 to $1000 overnight would be incredibly rare; much more rare than the same stock jumping from $1 to $1.05 overnight.  (This becomes important when dealing with something called the Efficient Market Hypothesis, which states that the market is a perfect synthesis of all available information about the value of a stock.  Therefore, the fluctuations would center around a price that was the inherently "correct" value.  I might deal with this more in a future post.)

Also important in his work was the implication that stock markets are inherently random.  After all, if they are  truly random, they follow statistical rules... and they can be predicted with statistical tools.  This was (and is) revolutionary.

It turns out that, unfortunately, Bachelier was wrong, in 2 important ways.  As Maury Osborne discovered in 1959, it is not the prices of stocks that are random, but the relative price changes in stocks.  This makes sense, since investors really care about the percent increase in the value of their stocks more than simply the change in price.  This problem is easily solved by describing the motion of the markets with geometric Brownian motion, which uses the log-normal distribution.  (Taking the logarithm of the difference in prices makes a Gaussian curve.)

Even Osborne didn't have the complete picture; Benoit Mandelbrot (mostly known for discovering fractal geometry) found the next piece of the puzzle, the one that has the biggest implication for modern econophysics.

If we look at the normal curve, we can see that statistical anomalies should not occur very often.  That is, price fluctuations should be relatively stable, with outliers occurring very rarely.  This is not the case in actual market dynamics.  Stocks over a single day typically change less than 2%; a movement of ten standard deviations therefore means a movement of at least 20%.  While Gaussian statistics tells us that such movements should happen once every 10^22 days (longer than the age of the universe) market data shows us that it happens at least once a week.  Something is clearly wrong with our model.

Mandelbrot, while analyzing cotton prices, found that they follow a distribution with fat tails.  Fat-tailed distributions are similar in shape to the normal curve, except that instead of approaching zero probability, their "tails" remain positive for a long time.

Probability of large fluctuations is greater in fat-tailed
distributions

This means, in essence, that improbable events in the markets happen a lot more often than we would imagine.  (Fat tails pop up in physics all the time, incidentally; I'll probably also deal with that in a future post.)

So Bachelier was essentially wrong in his most important argument-- that markets are like random walks. (To his credit, the data that he analyzed at the Paris Bourse did actually follow a random walk model-- perhaps it was especially stable at that time.)  However, he's still relevant to the field, because he established one of the first relationships between physics and the economy, fixed a precedent for analysis of statistical data in the economy, and argued that markets are random.  Economists mostly agree that he got that part right, and that since the markets are random, we can predict their (overall) behavior.  He also laid the groundwork for Mandelbrot to discover the market's fat tails, and thus put us one step closer to understanding some of the latest economic crises.

Sorry again for the pedantic rant, guys.  Have a great week, and feel free to comment!