Two physicists discovered the ultimate law of stock price fluctuations, and after publishing in a top journal, investors stayed up all night studying physics.

As data continues to grow, finance is increasingly resembling physics.

Written by | Wang Yu

Proofread by | Bu Zhou

Stock price prediction might be one of the most difficult mathematical problems in the world.

False and true news emerge endlessly, with bulls and bears engaging in a constant game of tug-of-war. Some believe that market fluctuations and other economic issues are completely unpredictable because we cannot know what every trader is thinking. Their thoughts can change at any moment with shifting economic conditions, leading to the conclusion that, in economics, no one can find laws as clear-cut as those in physics.

But just because we can’t predict each individual’s thoughts doesn’t mean economics lacks universal laws. Similar to physics, where even if we don’t understand the exact motion of each particle, physicists can still derive thermodynamic laws from their behavior. The renowned physicist Philip Anderson once said, “More is different.” Even without knowing the specific thoughts of individual traders, we can still abstract macro-level universal principles from the entire stock trading market.

Recently, two physicists from Kyoto University in Japan used data from the Tokyo Stock Exchange to actually discover a universal law governing the impact of buy and sell orders on stock prices. Their findings were published in the prestigious physics journal Physical Review Letters.

The rich data from the Tokyo Stock Exchange facilitated this research. Image source: Kakidai/Wikipedia

Physics-like Laws in Economics

Intuitively, it’s hard to imagine that economics would have “physical laws” that are objective, universal, and quantifiable, but in fact, many such laws do exist.

For example, trade volume between two countries often correlates with the distance between them—the farther apart, the lower the trade volume. Similarly, the product of their GDPs, akin to their masses, correlates with trade volume—the higher the product, the greater the trade.

In stock or commodity markets, the way prices change over time exhibits statistical behaviors similar to diffusion equations in physics.

Not to mention Benford’s Law, often used to detect data fraud in economics: in datasets spanning multiple orders of magnitude, the distribution of leading digits tends to follow a logarithmic pattern, with about 30% of the first digits being 1.

Benford’s Law for natural data digit distribution can be used to verify data authenticity. Image source: Gknor/Wikipedia

However, these laws are often too macro-level to be as tangible to individual traders as stock prices are. Interestingly, recent independent studies by multiple research teams have reported an empirical “square root law”: stock trading appears to influence stock prices in a manner proportional to the square root of trading volume.

How does trading affect stock prices? At least qualitatively, buying tends to raise the average price, while selling tends to lower it. If Q is the trading volume and I(Q) is the average price change, then based on the square root law, I(Q) is proportional to Qδ, where δ=1/2. However, many skeptics question whether this law, like the others mentioned, is a macro-level principle applicable to the entire market and not to individual stocks. In other words, they doubt whether the universality of the square root law is strong enough.

Complex Science

So far, it’s hard to say whether the 21st century is the century of biology, but it’s certain that it is the century of complex science. The explosive growth in computing power has given scientists unprecedented data analysis capabilities, turning vast, structured datasets into scientific goldmines.

Yuki Sato and Kiyoshi Kanazawa, physicists from Kyoto University, obtained an unprecedentedly rich dataset from the Tokyo Stock Exchange, validating the empirical square root law. This dataset includes all trading records over eight years, with each trade tagged to associate it with specific traders. Of course, the identities are anonymous—they could be large financial institutions or individual traders—but regardless of who they are, researchers can reconstruct trading sequences based on the tags, reflecting similar buying and selling intentions.

Previous studies on stock price volatility often suffered from limited data, forcing researchers to merge multiple stocks’ trading records, which could introduce additional noise. Thanks to this comprehensive dataset, researchers could test the square root law on individual stocks. The results were astonishing: the law proved to be both elegant and universal. It holds true for individual traders and for single stocks.

Buy orders (blue) and sell orders (red), as particles diffusing along the price axis (x), annihilate upon contact (yellow explosion), forming a V-shaped depletion layer near the contact point (trading price pt). Image source: L. Dall’Amico et al., J. Stat. Mech. 013404 (2019)

The researchers didn’t stop at verifying this law—they also sought to uncover its underlying cause. Inspired by reaction-diffusion systems in physics, they proposed a “latent liquidity” model. They first suggested that the available liquidity of a stock increases linearly with the current price’s distance, leading to the formation of a “depletion layer” of liquidity around the current price. This distribution naturally results in the emergence of the square root law.

However, setting aside this microscopic explanation, the law seems to emerge from the interactions of thousands of market participants—an emergent pattern. Such laws indicate that finance is gradually aligning with physics, possessing high-quality data, reproducible results—and outcomes that matter practically.

Researchers are extracting objective laws from complex systems like stock trading. They believe that these mechanisms could help understand the inner workings of financial markets, especially their incredible tendency to crash without warning.

Investing involves risks; please proceed cautiously. This article does not constitute financial advice.

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