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The Hidden Constant of Market Rhythms: How $1-1/e$ Defines Scaling in Intrinsic Time

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  • Thomas Houweling

Abstract

Directional-change Intrinsic Time analysis has long revealed scaling laws in market microstructure, but the origin of their stability remains elusive. This article presents evidence that Intrinsic Time can be modeled as a memoryless exponential hazard process. Empirically, the proportion of directional changes to total events stabilizes near $1 - 1/e = 0.632$, matching the probability that a Poisson process completes one mean interval. This constant provides a natural heuristic to identify scaling regimes across thresholds and supports an interpretation of market activity as a renewal process in intrinsic time.

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  • Thomas Houweling, 2025. "The Hidden Constant of Market Rhythms: How $1-1/e$ Defines Scaling in Intrinsic Time," Papers 2511.14408, arXiv.org.
    • Handle: RePEc:arx:papers:2511.14408
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    1. Ma, Junjun & Xiong, Xiong & He, Feng & Zhang, Wei, 2017. "Volatility measurement with directional change in Chinese stock market: Statistical property and investment strategy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 471(C), pages 169-180.
    2. Robert F. Engle & Jeffrey R. Russell, 1998. "Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data," Econometrica, Econometric Society, vol. 66(5), pages 1127-1162, September.
    3. V. Petrov & A. Golub & R. Olsen, 2020. "Agent-based modelling in directional-change intrinsic time," Quantitative Finance, Taylor & Francis Journals, vol. 20(3), pages 463-482, March.
    4. Vladimir Petrov & Anton Golub & Richard Olsen, 2019. "Instantaneous Volatility Seasonality of High-Frequency Markets in Directional-Change Intrinsic Time," JRFM, MDPI, vol. 12(2), pages 1-31, April.
    5. James B. Glattfelder & Thomas Houweling & Richard B. Olsen, 2025. "A Modern Paradigm for Algorithmic Trading," Papers 2501.06032, arXiv.org.
    6. Ole E. Barndorff‐Nielsen & Neil Shephard, 2002. "Econometric analysis of realized volatility and its use in estimating stochastic volatility models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(2), pages 253-280, May.
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