Beautiful Science: Molecules and Markets I

Wall Street workers are faced with a giant random number generator every day … You don’t have any control over the stock market, the economy, banking industry profits or legislation. You can lose money on your best ideas and make money on your worst ideas. Sometimes it seems totally random.

This quote is taken from a recent Bloomberg article by Jared Dillian, a former trader at Lehman Brothers Holding Inc., and currently an investment strategist at Mauldin Economics and author of several popular books on finance.

The gist of Dillian’s quote concerns a central feature of all financial markets: that the prices of traded assets, being buffeted by a multitude of factors, evolve randomly. To use the jargon, their prices are “noisy” signals of an asset’s intrinsic value. Noisiness exasperates many investors (most naturally those who are losing money) but at the same time, without noise, there is perfect agreement on asset values and hence, there will be no reason to trade. Markets would be boring, if they exist at all.

To students of probability (an important branch of mathematics), the randomness of markets isn’t that surprising. In fact, there is a surprising connection between the movement of molecules and the prices of traded financial assets. The root of this connection goes back to the 19th century in England. In 1827, the English biologist Robert Brown was looking under a microscope at pollen grains floating in water when he noticed something odd. Instead of gentle movements, he saw the pollen grains dart around erratically, like a drunk person staggering hither and thither. This random motion is christened Brownian motion in the scientific literature in honor of Dr. Brown. A simulation of Brownian motion (below) is shown below:

Brown was intrigued by what he saw but he couldn’t explain what was going on. Nor did he link pollen grains with financial markets. It wasn’t until 1905 that Albert Einstein came up with an explanation. Einstein proved, using partial differential or heat equations, that the erratic movements of the pollen grains was due to the underlying erratic movements of water molecules – millions upon millions of restless water molecules whizzing around.

More astonishingly, five years before Einstein’s breakthrough, an obscure Frenchman by the name of Louis Bachelier (1870–1946) was making a connection between Brownian motion and stock prices. Bachelier was a student of Henri Poincare, the renowned mathematician and theoretical physicist, and a polymath who excelled in all fields of the discipline as it existed during his lifetime. Under Poincare, Bachelier wrote a doctoral thesis on the theoretical properties of stock prices. It was in that thesis that Bachelier propose to model stock price movements as following a basic type of random (stochastic) process driven by Brownian motion. Indeed, Bachelier exceeded himself by deriving the price of what is now called a barrier option, a type of exotic derivative security. With that work, Bachelier essentially became the father of modern finance.

However, neither Bachelier nor Einstein provided a rigorous analysis of the properties of Brownian motions. This gap was filled in 1923 by the MIT mathematician Norbert Wiener (1894–1964). Iit was in honor of Wiener’s seminal contribution that the standard Brownian motion is named a Wiener process. Since that time, Brownian motion and its discrete-time cousin, the random walk, have found their way in innumerable applications not only in economics and finance but also in many branches of the physical sciences, an example of which is in the design of electronic, telecommunication and satellite systems.

Left: Louis Bachelier (1870 – 1946) whose1900 doctoral thesis introduced the first mathematical model of Brownian motion and its use for valuing stocks and stock options. Right: Albert Einstein in 1905 with his first wife and first-born son. Photo: American Institute of Physics. Einstein gave a physical explanation of the erratic motion of particles first observed by Robert Brown.
The famed American mathematician, Norbert Weiner (1894–1964) was the first to rigorously study the mathematical properties of Brownian motion as a form of random or stochastic process.

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