Financial markets are commonly considered chaotic; however, the physics of complex systems continues to uncover stable patterns within them. Japanese researchers Yuki Sato and Kiyoshi Kanazawa from Kyoto University have published empirical evidence of the so-called square-root law (Square-Root Law, SRL) using data from the Tokyo Stock Exchange in Physical Review Letters.
The essence of the law is simple and at the same time inconvenient for large players: the impact of a large trade on price grows not linearly, but proportionally to the square root of the trade volume. This means that to double the price impact, an investor must increase the volume of a buy or sell not twofold, but fourfold. In other words, the market “dampens” aggressive capital entries faster than it may seem at first glance.
Importantly, the study is based not on models or simulations, but on real data from one of the world’s largest exchanges. As a result, SRL strengthens its status as a universal law applicable not only to the Japanese market, but to global financial systems as a whole.
For investors and algorithmic trading, this has practical implications: large positions become an increasingly ineffective tool for influencing price, and market liquidity behaves like a physical medium with its own constraints. Financial markets remain risky—but increasingly appear less completely random.
Link to the publication: https://journals.aps.org/prl/abstract/10.1103/65jz-81kv
The square-root law is important not as an elegant theory, but as a working constraint under which modern markets already operate—especially where speed, liquidity, and algorithms decide everything. Its significance for HFT, market making, and crypto markets manifests differently, but everywhere it sets a boundary for capital efficiency.
High-Frequency Trading
For HFT strategies, the key objective is to extract micro-profits without moving the market against oneself. The square-root law shows that the more aggressive the volume, the faster one’s own price footprint grows. This means that increasing order size almost immediately reduces strategy efficiency: impact growth outpaces the growth of expected profit. In practical terms, this leads to order fragmentation, the use of execution algorithms (TWAP, VWAP, POV), and strict control of market footprint. HFT algorithms are optimized not for entry speed, but for minimizing impact cost—and SRL provides a quantitative framework for this.
Market Making
For market makers, the square-root law explains why order-book depth does not grow linearly, even in highly liquid markets. When a large participant enters the market, the market maker understands that the price impact will be sublinear, but inevitable. This directly affects spread dynamics, quoted volumes, and inventory risk management. Practically, SRL means that the higher the potential large flow, the faster a market maker reduces exposure or widens the spread, because price reacts not to intent, but to the mass of capital passing through the market.
Crypto Markets
In cryptocurrencies, the square-root law is especially visible. Despite external liquidity and round-the-clock trading, crypto markets have shallow depth, meaning SRL manifests more harshly and more quickly. Large purchases or sales do not lead to proportional price movements—instead, the market “breaks”: liquidation cascades, front-running, and jumpy volatility emerge. For large holders (whales), this means that direct entry or exit becomes expensive, and price impact grows nonlinearly very quickly. For DeFi and AMM models, SRL effectively explains why large swaps sharply worsen execution prices and why there is a need for aggregators and off-chain execution.
Overall conclusion
The square-root law shows that the market behaves like a physical medium with resistance. Capital cannot endlessly amplify its impact on price—each subsequent step becomes more expensive. For algorithms, this means volume optimization; for market makers, protection against imbalance; for crypto markets, structural volatility. And most importantly: the market punishes size, not speed or intelligence. That is why in modern trading, the winner is not the one who pushes the price harder, but the one who better understands how far they can go before the market starts pushing back.
