For years, researchers have debated whether financial markets follow universal laws like those in physics, with the square-root law (SRL) being one of the most contested ideas.
The square-root law suggests that the price impact of a trade rises with its size in a predictable way, specifically, proportional to the square root of the transaction volume.
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. Critics argue that this relationship should vary from market to market or even stock to stock. Supporters, however, believe the rule is universal.
The problem has never been the theory, it’s the data. Proving universality demands extremely precise measurements across hundreds of stocks, using detailed records that track the actions of individual traders. Those datasets are rare. This study on square-root law breaks that barrier by examining every trading account for all liquid stocks on the Tokyo Stock Exchange over an eight-year period.
The results are striking. Across individual stocks and individual traders alike, the price-impact exponent consistently lands at one-half, squarely in line with the square-root law. The analysis also rules out two major models that argue against universality, strengthening the case that the SRL holds across the board.
Beyond settling a long-running academic debate, the findings matter in the real world. They offer a sharper, more reliable way to estimate price impact, an issue that sits at the heart of how large investors trade. In short, markets may be more predictable than skeptics once thought.
The square-root law validated
The research on square-root law draws on an exceptionally rich microscopic dataset, allowing the authors to track the trading behavior of all individual accounts rather than relying on aggregated market data. This level of granularity makes it possible to measure the price-impact exponent with unprecedented precision, reducing statistical errors to levels that were previously out of reach.
Crucially, the study does not stop at confirming the square-root law. It also directly tests and rejects two prominent non-universality frameworks: the Gabaix-Gopikrishnan-Plerou-Stanley model and the Farmer-Gerig-Lillo-Waelbroeck model, both of which predict that price impact should vary across markets or assets.
Rep. Image credits: Freepik | Cropped by GBN
By delivering high-precision evidence at both the stock level and the trader level, the work stands out as one of the strongest empirical validations of a universal law in the social sciences. Its implications extend beyond theory, providing market participants with a more dependable foundation for assessing liquidity, execution costs, and the market impact of large trades.
The findings
Taken together, the findings suggest that financial markets may obey deeper structural laws similar to those found in the natural sciences. By demonstrating that a single, simple rule governs price impact across stocks, traders, and time, the study strengthens the case for universality in complex social systems.
In doing so, it not only reshapes how markets are understood, but also hints at a future where economic behavior can be modeled with greater confidence, precision, and predictability.
