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

ArXiv ID: 2511.14408 “View on arXiv”

Authors: 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.

Keywords: Directional-change Intrinsic Time, Scaling laws, Poisson process, Hazard process, Market microstructure, Equity / General Financial Markets

Complexity vs Empirical Score

  • Math Complexity: 7.0/10
  • Empirical Rigor: 5.0/10
  • Quadrant: Holy Grail
  • Why: The paper presents a sophisticated mathematical model using renewal theory and Poisson processes to derive a scaling constant, but it also implements a concrete empirical analysis on real cryptocurrency data with specific statistical tests and reproducibility references.
  flowchart TD
    A["Research Goal<br>Identify origin of stable<br>scaling laws in Intrinsic Time"] --> B["Key Methodology<br>Directional-change Intrinsic Time<br>Analysis & Poisson Process Modeling"]
    B --> C["Data Inputs<br>Empirical Intrinsic Time Series<br>Directional Changes & Total Events"]
    C --> D["Computational Process<br>Measure Ratio of Directional<br>Changes to Total Events"]
    D --> E{"Analysis"}
    E --> F["Evidence: Ratio stabilizes near<br><b>1 - 1/e = 0.632</b>"]
    E --> G["Interpretation<br>Market activity modeled as<br>Memoryless Exponential Hazard Process"]
    F --> H["Key Findings/Outcomes<br>1. Identified Hidden Constant<br>2. Natural heuristic for scaling regimes<br>3. Renewal Process interpretation"]
    G --> H