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

TL;DR

This paper shows that market activity in intrinsic time follows a memoryless exponential hazard process, with the proportion of directional changes stabilizing near 1 - 1/e, revealing a fundamental scaling law in market microstructure.

Contribution

It introduces a model of intrinsic time as a memoryless exponential hazard process, explaining the stability of scaling laws in market microstructure.

Findings

Proportion of directional changes stabilizes near 0.632

Market activity can be modeled as a renewal process in intrinsic time

Supports the interpretation of market dynamics as a Poisson process

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.