December 16, 2025 by Paul Arnold, Phys.org
Collected at: https://phys.org/news/2025-12-universal-law-large-stock-prices.html
Financial markets are often seen as chaotic and unpredictable. Every day, traders around the world buy shares and sell assets in a whirlwind of activity. It looks like a system of total randomness—but is it really?
Scientists have long suspected that there is a hidden order under this noise, but it has been difficult to prove. Now, Yuki Sato and Kiyoshi Kanazawa of Kyoto University have provided some of the strongest evidence yet. By studying eight years of data from the Tokyo Stock Exchange (TSE), they have confirmed a long-standing hypothesis known as the square-root law (SRL) of price impact.
What is the square-root law?
SRL is a mathematical formula that predicts how much a stock’s price will change after a large trade. It says that the price impact grows in proportion to the square root of the number of shares traded. So, for example, if a large investor buys a certain amount of stock and sees the price rise by 5%, doubling the order will not double the price move. Because of the square-root rule, they would have to buy four times as much stock to double the price impact.Square-root price impact law. Credit: Physical Review Letters (2025). DOI: 10.1103/65jz-81kv
For years, researchers have debated whether this kind of square-root relationship stays the same for everyone (a universal law) or changes depending on the specific stock, trader or market where it is traded. In a new study published in the journal Physical Review Letters, Sato and Kanazawa show that the rule is universal and applies to all markets, regardless of their individual details.
The study authors measured all order submissions, trades and cancellations on the Tokyo Stock Exchange for every single trading account from 2012 to 2019. The result was that, across all stocks and traders, the price impact followed the square-root rule.
“Our result supports the universality hypothesis of the SRL with extraordinary accuracy,” commented the researchers in their paper “The universality hypothesis holds for the SRL within statistical errors, at least for the TSE.”
Using the physics of finance
We are accustomed to universal laws in nature such as those governing gravity or thermodynamics, but finding such a precise rule for human behavior is rare. The research demonstrates that even though traders may be unpredictable, the market as a whole operates with the reliability of a physical system.
Beyond resolving a scientific debate, the SRL could be a tool to help investors plan trades in a way that reduces risks and allows regulators to better assess market stability during large-scale trading.
More information: Yuki Sato et al, Strict Universality of the Square-Root Law in Price Impact across Stocks: A Complete Survey of the Tokyo Stock Exchange, Physical Review Letters (2025). DOI: 10.1103/65jz-81kv. On arXiv: DOI: 10.48550/arxiv.2411.13965
Journal information: Physical Review Letters , arXiv
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