Pollen, Stock Values, and Brownian Motion

Brown noticed the phenomenon that would eventually come to be called “Brownian motion” while studying the pollen of this plant, Clarkia pulchella.

Image credit: Public domain.

Image Credit: K Kanazawa, T. Sueshige, H. Takayasu, and M. Takayasu.

Unfortunately, no matter how much you know about a stock, you still can’t know for sure how its price will change next. In the same way, no matter how much you know about a coin before it’s flipped, you still can’t predict which face it will land on next. The common factor? Randomness.In math and physics,means basically the same thing as in daily conversation—unpredictable. A system is random when you can’t predict an outcome with certainty, even when you have all of the available information. But you can explore the behavior of random systems using mathematical models, which operate in the world of probabilities, describing thebehavior of random systems under different conditions.In order to better understand randomness in the financial markets, a team of scientists from Tokyo Institute of Technology recently analyzed the decisions of high-frequency foreign exchange traders. Based on their results, the researchers developed a theory that explains the randomness in financial markets that’s baffled economists for more than 100 years. The research was published this week in the American Physical Society’s journalFirst, some background. In 1827, botanist Robert Brown turned his microscope on the pollen of a wildflower. He noticed something curious: tiny particles, suspended in the water that contained the pollen, jittering around. Brown went on to show that the random movements weren’t signs of life, but he could never figure out the cause.Nearly 80 years later, Albert Einstein explained the source of the jitters. In groundbreaking research, he laid the framework for the kinetic theory of matter—the idea that stuff is made of atoms and molecules that are constantly in motion. His paper showed that when water molecules collide with tiny particles suspended in a liquid, the particles jitter and move, just as Brown had described. This was a convincing argument for the existence of atoms and molecules, a matter still under debate at that time.Five years before Einstein’s paper, the French mathematician Louis Jean-Baptiste Alphonse Bachelier published his PhD thesis. A pioneer even in his graduate years, he was the first person to mathematically model how random behavior evolves over time. Then he applied this to the stock market, another first, and showed how it could be used to value stock options. Brownian motion is the term now used to describe how the location of particles suspended in a liquid evolve over time, and more generally how any system with random fluctuations evolves over time. The same mathematical model Bachelier applied to stocks also applies to Brown’s suspended pollen grains, along with many other situations. For a home demonstration of Brownian motion and kinetic theory, put one drop of food coloring in a glass of cold water, and one in a glass of hot, and watch how quickly each disperses.The fact that physical Brownian motion parallels financial Brownian motion caught the attention of physicists early on. The subject has been studied often, but a key link has been missing: Einstein’s 1905 paper explained the microscopic cause of the random motion of the particles—water molecules in motion—but the analogous cause of Brownian motion in stock prices hasn’t been clear until now.In previous research, the team at Tokyo Institute of Technology showed that financial markets exhibit similarities to physical Brownian motion in the way price changes with time, and also in the number of shares of a stock requested or offered at different prices (called the book-order). Based on this, they suspected that the behavior of individual traders drives financial Brownian motion.In this new research, led by Misako Takayasu, the team analyzed a week’s worth of foreign exchange data between the US dollar and Japanese Yen. They focused their attention on the behavior of high frequency traders, looking for trends.High frequency traders use computer-based algorithms to determine the best price to buy or sell a financial product at. Let’s say a trader submits an order to sell something at a certain price. If there isn’t a match between the trader’s price and what a buyer wants to pay, the transaction doesn’t occur and the order is eventually canceled. The seller then modifies the price according to an algorithm and submits another order. If another trader agrees to the new price, a transaction takes place, otherwise the order is canceled and the processes repeats. Automate this process so that it happens repeatedly within a matter of seconds, and you’ve got today’s high frequency traders.The team found that high frequency traders made decisions about the price at which to sell or buy based on what’s called a “trend-following” strategy—if the price went up previously, the traders expected the price to go up again. If the price went down, the traders expected the price to go down again. These predictions are based on a concept that shows up a lot in statistics, called regression toward the mean. For long trends, this behavior eventually reached saturation.Using the characteristics of this trend-following behavior they uncovered, the researchers constructed a kinetic theory of financial Brownian motion, analogous to the kinetic theory that explains physical Brownian motion. This theory, they show, leads to exactly the price-time and book-order behaviors of financial systems that have been previously observed.In other words, the “moving molecules” in financial Brownian motion are the trend-following decisions of individual traders.Knowing this won’t help you hit it big in the stock market (sorry!), but it does put financial activity in a broader context. It offers a systematic explanation of financial Brownian motion from the microscopic scale (the decisions of individual traders) to the macroscopic (the change in price over time). It provides a foundation for understanding price fluctuations in stable markets, says Misako, and a jumping-off point for investigating unstable markets under external shocks.The story of financial Brownian motion demonstrates the power of applying the rigorous, scientific approaches to problems in other fields. “We believe that our research is a good example of how the methodology of physics works well even for social phenomena,” says Misako. She continues, “We hope that physicists can contribute to other fields where sufficient datasets are available.”