Why do financial prices exhibit Brownian motion despite predictable order flow?
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916761138364416 |
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| author | Sato, Yuki Kanazawa, Kiyoshi |
| author_facet | Sato, Yuki Kanazawa, Kiyoshi |
| contents | In financial market microstructure, there are two enigmatic empirical laws: (i) the market-order flow has predictable persistence due to metaorder splitters by institutional investors, well formulated as the Lillo-Mike-Farmer model. However, this phenomenon seems paradoxical given the diffusive and unpredictable price dynamics; (ii) the price impact $I(Q)$ of a large metaorder $Q$ follows the square-root law, $I(Q)\propto \sqrt{Q}$. Here we theoretically reveal why price dynamics follows Brownian motion despite predictable order flow by unifying these enigmas. We generalize the Lillo-Mike-Farmer model to nonlinear price-impact dynamics, which is mapped to an exactly solvable Lévy-walk model. Our exact solution shows that the price dynamics remains diffusive under the square-root law, even under persistent order flow. This work illustrates the crucial role of the square-root law in mitigating large price movements by large metaorders, thereby leading to the Brownian price dynamics, consistently with the efficient market hypothesis over long timescales. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_17906 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Why do financial prices exhibit Brownian motion despite predictable order flow? Sato, Yuki Kanazawa, Kiyoshi Trading and Market Microstructure Statistical Mechanics General Economics Economics Mathematical Finance Pricing of Securities In financial market microstructure, there are two enigmatic empirical laws: (i) the market-order flow has predictable persistence due to metaorder splitters by institutional investors, well formulated as the Lillo-Mike-Farmer model. However, this phenomenon seems paradoxical given the diffusive and unpredictable price dynamics; (ii) the price impact $I(Q)$ of a large metaorder $Q$ follows the square-root law, $I(Q)\propto \sqrt{Q}$. Here we theoretically reveal why price dynamics follows Brownian motion despite predictable order flow by unifying these enigmas. We generalize the Lillo-Mike-Farmer model to nonlinear price-impact dynamics, which is mapped to an exactly solvable Lévy-walk model. Our exact solution shows that the price dynamics remains diffusive under the square-root law, even under persistent order flow. This work illustrates the crucial role of the square-root law in mitigating large price movements by large metaorders, thereby leading to the Brownian price dynamics, consistently with the efficient market hypothesis over long timescales. |
| title | Why do financial prices exhibit Brownian motion despite predictable order flow? |
| topic | Trading and Market Microstructure Statistical Mechanics General Economics Economics Mathematical Finance Pricing of Securities |
| url | https://arxiv.org/abs/2502.17906 |