The Universal Law Behind Market Price Swings

Jean-Philippe Bouchaud

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The Universal Law Behind Market Price Swings

Capital Fund Management, Paris, France

December 15, 2025• Physics 18, 196

Analysis of a large dataset from the Tokyo Stock Exchange validates a universal power law relating the price of a traded stock to the traded volume.

Figure caption — **Figure 1:** Using a large dataset on stocks traded at the Tokyo Stock Exchange, Sato and Kanazawa have provided evidence of a universal “square-root law” describing financial markets [2].

One often hears that economics is fundamentally different from physics because human behavior is unpredictable and the economic world is constantly changing, making genuine “laws” impossible to establish. In this view, markets are never in a stable state where immutable laws could take hold. I beg to differ. The motion of particles is also unpredictable, and many physical systems operate far from equilibrium. Yet, as Phil Anderson argued in a seminal paper [1], universal laws can still emerge at the macroscale from the aggregation of widely diverse microscopic behaviors. Examples include not only crowds in stadiums or cars on highways but also economic agents in markets.

Now Yuki Sato and Kiyoshi Kanazawa of Kyoto University in Japan have provided compelling evidence that one such universal law governs financial markets. Using an unprecedentedly detailed dataset from the Tokyo Stock Exchange, they found that a single mathematical law describes how the price of every traded stock responds to trading volume [2] (Fig. 1). The result is a striking validation of physics-inspired approaches to social sciences, and it might have far-reaching implications for how we understand market dynamics.

Several robust quantitative laws have already been empirically identified in economics and finance. Among the best known are the “gravity law” that relates the trade between countries to their economic size and geographic distance; the diffusion-like laws that describe how market prices evolve in time [3]; and the power law observed in distributions of wealth, income, city sizes, and firm sizes [4]. More recently, several research groups have independently reported an empirical “square-root law” that appears to govern price impact—the effect of the trade of an asset on the price of that asset. Hints of such a law have been documented across a wide variety of financial markets (for a review, see [3]).

The law quantifies the response of the market to a perturbation (a buy or a sell order). To understand the law’s meaning, consider an investment firm that wants to buy a quantity $Q$ of some asset: stocks, crude oil, currency, or even bitcoin. How much will this order move the price, on average? During the execution of the trade, prices will fluctuate in the diffusive manner mentioned earlier, drifting up and down depending on the time it takes to complete the trade. But when one averages over a very large number of trades of the same size $Q$ , clear patterns emerge. First, the average price change is positive for buy orders and negative for sell orders. Second, the magnitude of the change is proportional to two factors: the “volatility” of the asset (its propensity to fluctuate, or, in physics terms, its diffusion constant) and the square root of the ratio of the volume $Q$ to the total volume traded by the whole market over the same period.

The law is reminiscent of Einstein’s fluctuation–dissipation relation, which states that the displacement of a Brownian particle subject to an external force is proportional to the strength of the perturbation and to the diffusion constant in the absence of that perturbation. But the analogy isn’t exact. Whereas Einstein’s law describes a linear response of a particle to the force, the response of price to a perturbation is nonlinear: Buying twice as much does not move the price by twice as much, but only by a factor of $\sqrt{2}$ . An intriguing implication of a square-root dependence is that the marginal response to vanishing perturbation is infinite (as the derivative of $\sqrt{Q}$ diverges at $Q = 0$ ).

This behavior is highly intriguing, but a nagging concern remains: Is this square-root dependence a truly universal law? Or could it be an artefact, arising from subtle conditioning effects (for example, buyers reducing their trading as price goes up), from biased samples (such as ones containing only trades by large financial institutions) or from other quirks of the used datasets? Much of the data used in earlier studies are proprietary and not publicly available, which has prevented independent verifications. Those data also covered only a tiny fraction of all transactions. Before elevating the square-root dependence to the status of a bona fide law of financial nature, far more comprehensive evidence is needed.

The work of Sato and Kanazawa provides exactly this kind of confirmation. The duo gained access to a unique database from the Tokyo Stock Exchange, containing all transactions over an eight-year period, with each trade tagged so that it can be associated with a unique—though anonymous—decision-maker. This tagging allowed the researchers to reconstruct a complete sequence of trades stemming from the same underlying buy or sell intention, whether originating from a major financial institution, a retail trader, or a market maker.

Crucially, the dataset is rich enough to test the square-root law stock by stock, overcoming the limitations of past analyses that had to pool data across many different stocks to reduce noise. The result is striking: The square-root law holds beautifully and universally, regardless of stock-specific features previously thought to shape the function describing price impact (such as the distribution of traded volumes $Q$ or the number of “child orders” that make up larger “parent orders”).

What microscopic behavior could account for the emergence of the square-root law? The analysis of Sato and Kanazawa is sufficiently precise to rule out two previously proposed mechanisms, which implied a nonuniversal price impact [5, 6]. My collaborators and I have proposed a viable framework, which we call “latent liquidity” [3, 7, 8]. Inspired by reaction–diffusion systems in physics [9], this model is built on the idea that available liquidity (the sell volume that can match an incoming buy order or vice versa) increases linearly with the distance from the current price (Fig. 2). This is not an ad hoc assumption but a mathematical consequence of price diffusion itself, which creates a liquidity “depletion layer” around the current price, much like diffusive reactants are depleted close to the reaction zone (Fig. 2). The concave shape of the square-root scaling can then be visualized as the outcome of a linearly growing liquidity profile that increasingly resists further price moves. This model accounts well for many empirical observations beyond the square-root law itself but isn’t yet fully consistent with data. Datasets as rich and comprehensive as those analyzed by Sato and Kanazawa will provide fertile grounds for these ideas.

Independent of its microscopic interpretation, the square-root law appears to be a genuine emergent law of supply and demand, resulting from the interaction of thousands of market participants. Results like this show that empirical finance is approaching the standards of experimental physics: high-quality data, reproducible results, and the ability to test nontrivial predictions with high precision. For price impact, the stakes are high: It is the key channel through which decisions are converted into price changes and lies at the heart of recent, heated debates about the efficiency and fragility of financial markets [10, 11]. The confirmation of the square-root law may shed light on the inner workings of financial markets, in particular, on their uncanny propensity to crash without warning.

References

P. W. Anderson, “More is different,” Science 177, 393 (1972).
Y. Sato and K. Kanazawa, “Strict universality of the square-root law in price impact across stocks: A complete survey of the Tokyo stock exchange,” Phys. Rev. Lett. 135, 257401 (2025).
J.P. Bouchaud et al., Trades, Quotes and Prices: Financial Markets Under the Microscope (Cambridge University Press, Cambridge, 2018)[Amazon][WorldCat].
X. Gabaix, “Power laws in economics and finance,” Annu. Rev. Econ. 1, 255 (2009).
X. Gabaix et al., “Institutional investors and stock market volatility,” Q. J. Econ. 121, 461 (2006).
J. D. Farmer et al., “How efficiency shapes market impact,” Quant. Finance 13, 1743 (2013).
I. Mastromatteo et al., “Agent-based models for latent liquidity and concave price impact,” Phys. Rev. E 89, 042805 (2014).
J. Donier et al., “A fully consistent, minimal model for non-linear market impact,” Quant. Finance 15, 1109 (2015).
G. T. Barkema et al., “Reaction-diffusion front for $A + B \to \emptyset$ in one dimension,” Phys. Rev. E 53, R2017 (1996).
X. Gabaix and R. S. Koijen, “In search of the origins of financial fluctuations: The inelastic markets hypothesis,” NBER Working Paper No. 28967 (2021).
J. P. Bouchaud, “The inelastic market hypothesis: A microstructural interpretation,” Quant. Finance 22, 1785 (2022).

About the Author

Jean-Philippe Bouchaud is the chairman of Capital Fund Management (CFM) and a professor at École Normale Supérieure (ENS), both in France. With Jean-Pierre Aguilar, he cofounded CFM’s research arm, Science and Finance, in 1994 and maintains strong links with the academic world. Prior to CFM, he was a researcher at the French National Centre for Scientific Research (CNRS) until 1992. He then spent a year at the Cavendish Laboratory at the University of Cambridge in the UK before joining the Condensed Matter Physics Service at the French Atomic Energy and Alternative Energies Commission. He holds a PhD in theoretical physics from the ENS.